Our main theorem gives a dichotomy: every finite orbit is either of pullback type (projectively factoring through an orbifold curve, hence arising from classical branched-cover constructions), or of Katz–reflection type, meaning it is obtained from a finite complex reflection group in dimension n − 2 via a bounded sequence of Katz operations (rank-one twists and middle convolution). For n = 5, 6 we prove that only finitely many reflection-group data can occur and we enumerate them explicitly (using the Shephard–Todd classification plus certified braid-action computations in trace/cluster coordinates). The result produces a complete atlas of finite braid orbits in rank two for n = 5, 6, providing a rigorous testbed for arithmetic conjectures on algebraic isomonodromy and non-linear analogues of p-curvature.
Let X = ℙ1 \ {x1, …, xn}
be an n-punctured sphere with
ordered punctures, where throughout n ∈ {5, 6}. We study the action of
the pure mapping class group PMod0, n ≅ π1(ℳ0, n)
on the rank-two Betti moduli space
Y(0, n, 2) = Hom(π1(X), SL2(ℂ))/ /SL2(ℂ),
and in particular on its irreducible locus Y(0, n, 2)irr. A
point [ρ] ∈ Y(0, n, 2)
determines a conjugacy class of tuples (A1, …, An) ∈ SL2(ℂ)n
with ∏iAi = I,
modulo simultaneous conjugation, and the PMod0, n-action is the
geometric incarnation of the braid/Hurwitz dynamics on such tuples
induced by outer automorphisms of π1(X). Our goal
is to classify those [ρ] ∈ Y(0, n, 2)irr
whose PMod0, n-orbit is
finite.
The motivation comes from isomonodromy. Fixing local monodromy conjugacy classes C = (C1, …, Cn) and varying the puncture positions (x1, …, xn) yields an isomonodromic deformation problem whose nonlinear equations are the Schlesinger system in general, and the Garnier systems for n ≥ 5. Finite PMod0, n-orbits provide algebraic leaves of the isomonodromy foliation: moving around loops in ℳ0, n permutes the monodromy data by PMod0, n, hence a finite orbit forces the analytic continuation of the corresponding isomonodromic solution to take only finitely many values in the character variety. In the familiar n = 4 case, this philosophy underlies the relation between finite braid orbits and algebraic solutions of Painlev'e~VI; for n = 5, 6 it governs algebraic solutions of the relevant Garnier systems. We thus view finite orbits as the Betti-side avatar of a rigidity phenomenon for nonlinear differential equations.
The case n = 4 is qualitatively different in two ways. First, the moduli space Y(0, 4, 2) is a (typically affine) surface, and the braid group action is already rich enough that many approaches exploit explicit coordinates (Fricke–Klein or trace coordinates) and the detailed birational dynamics. Second, the landscape of finite orbits for n = 4 has been intensively studied: one sees both pullback constructions (originating from triangle groups and hypergeometric equations) and a finite supply of exceptional orbits related to the Platonic groups, often organized by explicit combinatorics of the braid action. Our treatment for n = 5, 6 is not a direct generalization of the n = 4 coordinate dynamics; instead, it is organized around a structural dichotomy guided by fixed-locus geometry and Katz functoriality.
A further difference from some earlier work is that we impose no restriction on the orders of the local monodromies Ai. Several classification results in the literature become more tractable under an ``infinite-order hypothesis’’ (for instance, requiring at least one local monodromy to have infinite order, or excluding special eigenvalue configurations) because such hypotheses can preclude large finite monodromy groups or eliminate certain resonance phenomena in Katz operations. In our setting, allowing all Ai to be of finite order is essential: it produces genuinely new finite PMod0, n-orbits, particularly in the n = 6 case, and it forces us to treat more systematically the interplay between braid dynamics, local monodromy strata Y(C), and the behavior of eigenvalues under middle convolution and twisting. The price is that the classification cannot rest on genericity assumptions; instead, we isolate the required finiteness and boundedness statements as explicit lemmas, and we separate the computer-verified components from the conceptual argument.
Our main theorem establishes a dichotomy for finite PMod0, n-orbits in Y(0, n, 2)irr for n = 5, 6. The first alternative is geometric: a finite orbit may arise from a pullback along a branched covering g : ℙ1 → ℙ1 of a projective local system on ℙ1 \ {0, 1, ∞}, equivalently from a projective representation factoring through an orbifold curve. This mechanism is the natural generalization of the classical hypergeometric pullback constructions and produces, in families, finite orbits that are typically non-isolated in character varieties. The second alternative is functorial and finite-group-theoretic: up to invertible Katz operations (rank-one twists and middle convolution/inverse), the local system reduces to one whose monodromy is a finite complex reflection group in dimension n − 2, generated by pseudoreflections with prescribed local conjugacy classes. In this second case the orbit points are rigid in the sense that their PMod0, n-stabilizers fix them in a zero-dimensional locus; this is the regime in which one expects a finite explicit list rather than continuous families.
The conceptual pivot is therefore the distinction between isolated and non-isolated finite orbit points. On the one hand, if a finite orbit point lies on a positive-dimensional fixed locus for a finite-index subgroup of PMod0, n, then the phenomenon is governed by the existence of nontrivial deformations compatible with isomonodromy symmetry. In this situation we invoke the Corlette–Simpson and Loray–Pereira–Touzet pullback criterion in the form adapted to punctured spheres: non-isolated behavior forces the associated projective local system to descend to an orbifold curve, and hence to arise (after normalization) from ℙ1 \ {0, 1, ∞} via pullback. On the other hand, if the point is isolated, then the same absence of deformations can be turned into constraints on how Katz operations may transform the system. For n = 5, 6 these constraints yield a bounded reduction statement: after twisting by a finite-order rank-one local system and applying a suitably chosen middle convolution MCλ−1 with λ ≠ 1 a root of unity, we obtain a local system with finite monodromy whose local monodromies at n − 1 punctures are pseudoreflections in GLn − 2(ℂ).
At this point, the finiteness of the classification becomes accessible because n − 2 ∈ {3, 4}. Finite irreducible complex reflection groups in dimensions 3 and 4 are completely classified (Shephard–Todd), and only finitely many admit generating tuples of pseudoreflections compatible with the determinant and product constraints inherited from rank 2 via Katz operations. This produces a finite collection of ``Katz–reflection data’’ (G, λ, D) serving as seeds. From each seed we reconstruct candidates in Y(0, n, 2) by applying explicit inverse Katz chains (a bounded number of twists and middle convolutions), and we then test whether the resulting tuples indeed yield finite PMod0, n-orbits in the irreducible locus.
The outcome is not only a qualitative dichotomy but an effective classification for n = 5 and n = 6. We produce two explicit finite lists 𝒫n and ℒn: the former consists of pullback data (orbifold monodromy together with a finite set of branch passports for g), and the latter consists of Katz–reflection data. The pullback list is finite by a Riemann–Hurwitz argument: since n is small, the possible ramification patterns over {0, 1, ∞} and the number of additional branch points compatible with producing exactly n punctures (up to the allowed local monodromy divisibility constraints) are bounded. The reflection list is finite by the preceding reduction to a bounded region of the Shephard–Todd table together with explicit determinant/product filters. Both lists are effectively computable.
A methodological point is that the reflection side necessarily involves computation. Even once one has an explicit tuple $(A_1,\dots,A_n)\in \mathrm{SL}_2(\overline{\mathbb{Q}})^n$ with ∏iAi = I, finiteness of the PMod0, n-orbit is a dynamical statement about the closure of the tuple under Hurwitz moves (or under a finite generating set of PMod0, n after choosing a standard identification). We therefore formalize a ``certified enumeration’’ protocol: for each candidate we record matrices over a number field, the explicit words implementing the generators of the relevant braid/mapping class action, and a finite set S certified to be closed under those generators; we also certify irreducibility. This isolates the computer-dependent component into explicit lemmas and allows the remainder of the argument to proceed in a conventional proof-theoretic manner.
Finally, we emphasize what genuinely changes when the infinite-order hypothesis is dropped. When all local monodromies are allowed to be finite order, one encounters finite-orbit points whose monodromy groups are finite but not captured by naive pullback constructions; in particular, Katz operations may pass through intermediate systems with special eigenvalue configurations, and one must control how the eigenvalues evolve to ensure the reduction terminates in bounded length. The bounded Katz reduction statement is precisely the mechanism that restores finiteness: for n = 5, 6 isolation forces the Katz chain into a bounded search space, and the Shephard–Todd classification then completes the reduction. The net effect is that the classification remains finite and explicit even in the presence of these additional finite-order phenomena.
In the next section we recall the basic geometry of Y(0, n, 2), the role of local monodromy strata Y(C), and the mapping class and braid group actions. This background provides the language for the fixed-locus alternative and the Katz-reduction arguments that drive the proofs of the main theorems.
We fix once and for all a basepoint x0 ∈ X and
simple positively oriented loops γi around xi based at
x0 such that
π1(X, x0) ≅ ⟨γ1, …, γn ∣ γ1⋯γn = 1⟩.
A representation ρ : π1(X) → SL2(ℂ)
is then equivalent to a tuple (A1, …, An) ∈ SL2(ℂ)n
with Ai = ρ(γi)
and $\prod_{i=1}^n A_i=I$. We
write
$$
R_n:=\Bigl\{(A_1,\dots,A_n)\in \mathrm{SL}_2(\mathbb{C})^n \
\Bigm|\ A_1\cdots A_n=I\Bigr\},
$$
an affine variety cut out by the matrix equation A1⋯An = I.
The group SL2(ℂ) acts on
Rn by
diagonal conjugation,
g ⋅ (A1, …, An) = (gA1g−1, …, gAng−1),
corresponding to changing the framing of the associated local system at
the basepoint.
The SL2-character variety
is the affine GIT quotient
Y(0, n, 2) := Rn/ /SL2(ℂ) = Spec(ℂ[Rn]SL2).
By standard GIT, its closed points correspond to closed SL2-orbits in Rn, equivalently
to polystable representations, i.e. semisimple rank-two local systems on
X. Concretely, two tuples
(Ai) and
(Ai′)
define the same point of Y(0, n, 2) if and only if
the corresponding representations have the same character (the same
values of tr(ρ(γ))
for all γ ∈ π1(X));
equivalently, their orbit closures in Rn intersect. In
particular, a reducible representation (one preserving a line in ℂ2) determines the same point as
its semisimplification (diagonal part), and this is the mechanism by
which non-closed conjugacy orbits are collapsed by the quotient.
We recall the basic notion of irreducibility. A representation ρ is if ℂ2 has no nontrivial ρ(π1)-invariant
subspace, equivalently if the matrices A1, …, An
are not simultaneously conjugate into the standard Borel subgroup of
upper triangular matrices. We denote by Rn, irr ⊂ Rn
the Zariski-open locus of irreducible tuples, and by Y(0, n, 2)irr ⊂ Y(0, n, 2)
its image. For SL2 one has
the useful simplification that an irreducible conjugacy orbit in Rn is
automatically closed, and its stabilizer is the center {±I}. Thus on Rn, irr the GIT
quotient is, up to the finite central ambiguity, the naive orbit space;
in particular, issues of semisimplification arise only on the reducible
locus. From the dimension count dim SL2 = 3, the relation A1⋯An = I
cuts dimension 3, and the quotient
removes dimension 3, we obtain the
expected dimension
dim Y(0, n, 2) = 3n − 6,
at least on the irreducible part where the quotient behaves well.
Local monodromy constraints are imposed by fixing conjugacy classes.
Let C = (C1, …, Cn)
be a choice of conjugacy classes in SL2(ℂ). We define the
$$
R(\mathbf{C}):=\Bigl\{(A_1,\dots,A_n)\in R_n \ \Bigm|\ A_i\in C_i\
\text{for all }i\Bigr\},
$$
and the corresponding
Y(C) := R(C)/ /SL2(ℂ) ⊂ Y(0, n, 2).
Since a conjugacy class in SL2(ℂ) is typically determined by
the trace (and in any case has codimension 1 in SL2 away from tr = ±2), one expects
dim Y(C) = 2n − 6
for generic C. For
n = 5 (resp. n = 6) this gives relative dimension
4 (resp. 6), which is the natural phase-space
dimension for the corresponding Garnier systems. We emphasize that we
work uniformly across all local conjugacy types, including the resonant
cases tr(Ai) = ±2,
where Ci
may be unipotent or semisimple and the geometry of Y(C) may acquire
singularities or multiple components. When necessary we will restrict to
the irreducible locus
Y(C)irr := Y(C) ∩ Y(0, n, 2)irr,
which is Zariski-open in the union of components of Y(C) that contain
irreducible representations.
It is often convenient to encode Ci by
eigenvalues. If A ∈ SL2(ℂ) is semisimple
with eigenvalues (α, α−1), then
its conjugacy class is determined by α + α−1 = tr(A)
up to exchanging α ↔︎ α−1. In the
unipotent case tr(A) = ±2 one
must distinguish the central elements ±I from nontrivial unipotents. We
will therefore treat C
as a genuine tuple of conjugacy classes, while keeping in mind that in
the generic semisimple regime it is equivalent to fixing the trace
parameters
ti := tr(Ai) ∈ ℂ, i = 1, …, n.
The trace functions are SL2-invariant regular functions on
Rn, hence
descend to regular functions on Y(0, n, 2); in particular
there is a natural morphism Y(0, n, 2) → 𝔸n
recording (t1, …, tn),
and Y(C) is a
(possibly non-reduced) fiber of this morphism when the Ci are specified
by trace data.
We next recall the mapping class and braid group actions in a form compatible with our tuple model. The group Mod0, n acts on X (up to isotopy), hence on π1(X) by outer automorphisms, and therefore on Hom(π1(X), SL2) by precomposition. Since simultaneous conjugation of a representation corresponds to changing the identification of the fiber at the basepoint, this action descends to an algebraic action on the character variety Y(0, n, 2). The pure mapping class group PMod0, n fixes each puncture xi pointwise; on the level of π1 it preserves the conjugacy class of each peripheral loop γi, so its action preserves every local monodromy stratum Y(C).
On tuples (A1, …, An)
with A1⋯An = I,
the same dynamics is described by Hurwitz (braid) moves. Let Bn be the Artin
braid group on n strands with
standard generators σ1, …, σn − 1.
The (right) Hurwitz action of Bn on Rn is given
by
σi : (A1, …, Ai, Ai + 1, …, An) ↦ (A1, …, Ai + 1, Ai + 1−1AiAi + 1, …, An),
which preserves the product relation and commutes with simultaneous
conjugation. The inverse move is
σi−1 : (A1, …, Ai, Ai + 1, …, An) ↦ (A1, …, AiAi + 1Ai−1, Ai, …, An).
Geometrically, these correspond to changing the generating system of
π1(X)
induced by dragging punctures around one another. In particular, Hurwitz
moves permute the local monodromies up to conjugation, and the full
braid group action does not preserve a fixed ordered tuple of conjugacy
classes (C1, …, Cn)
unless one allows the corresponding permutation of indices. By contrast,
the subgroup corresponding to PMod0, n (pure braids on
the sphere, equivalently mapping classes fixing punctures) acts without
permuting the punctures, and therefore preserves Y(C) for a fixed
ordered C.
Finally, we note that the distinction between ordered and unordered punctures is reflected in whether we work with PMod0, n or Mod0, n. The quotient Mod0, n/PMod0, n is naturally a subgroup of the symmetric group on n letters, acting by permutation of punctures; on the tuple side, this is realized by permuting the factors (A1, …, An). In our classification statements we keep the punctures ordered, so that the stabilizer and orbit notions are unambiguous on each stratum Y(C), and we treat permutation of punctures as an additional equivalence when passing between pure and full mapping class group orbits.
The constructions above provide the basic framework in which our finiteness problem lives: points of Y(0, n, 2) are characters of representations of π1(X), the subvarieties Y(C) encode prescribed local monodromy, and PMod0, n acts by algebraic automorphisms induced by Hurwitz dynamics. In the next section we will introduce explicit coordinate systems (trace and cluster-type charts) in which these actions become concrete birational transformations for n = 5, 6.
For the concrete study of PMod0, n-dynamics we will use two complementary atlases on Y(0, n, 2) and on its local monodromy strata: (i) trace (Fricke–Vogt) coordinates, which are intrinsic and extend across resonant local conjugacy types, and (ii) cluster-type coordinates coming from ideal triangulations, in which mapping classes act by explicit birational ``mutation’’ formulas on a Zariski-open set. We emphasize that we do not seek a single global polynomial presentation of Y(0, n, 2); rather, we isolate coordinate systems adapted to computations and to the detection of finite orbits.
Let Γ = π1(X, x0)
and ρ : Γ → SL2(ℂ).
For any word w ∈ Γ we
consider the regular class function
trw([ρ]) := tr(ρ(w)) ∈ ℂ.
The coordinate ring ℂ[Y(0, n, 2)] is generated
by such trace functions; moreover, for SL2 there are effective finite
generating sets. We will repeatedly use the basic skein identity: for
any M, N ∈ SL2(ℂ),
together with cyclicity tr(UVW) = tr(WUV)
and tr(M) = tr(M−1).
These allow us to rewrite traces of longer words as polynomials in
traces of shorter ones, once a fixed generating set is chosen.
In our low-dimensional cases n ∈ {5, 6}, it is convenient to
eliminate An using the
relation A1⋯An = I,
so that Γ is generated by
n − 1 elements with no further
relations. By the classical Fricke–Vogt–Procesi theory for SL2, the SL2-invariants of a free group
representation are generated by traces of words of length ≤ 3 in the chosen generators. Concretely,
writing Bi := Ai
for i ≤ n − 1, a
convenient generating family on Y(0, n, 2) is
ti := tr(Bi), tij := tr(BiBj), tijk := tr(BiBjBk),
for 1 ≤ i < j < k ≤ n − 1.
For n = 5 (free rank 4) and n = 6 (free rank 5), this gives explicit affine embeddings
into a space of moderate dimension, together with explicit defining
relations. While we will not record these relations in full, we stress
that every trace of a word appearing in a Hurwitz transform can be
reduced algorithmically to a polynomial in the chosen generators by
repeated application of . This is the mechanism by which we make the
braid/mapping class action explicit in trace coordinates.
On a local monodromy stratum Y(C) the functions ti = tr(Ai) are fixed (when Ci is determined by trace), and the effective phase-space dimension is 2n − 6. In practice we work with trace generators that separate generic points on Y(C)irr and are adapted to a pants decomposition of the punctured sphere. For example, for n = 5 we may take the traces of the boundary loops together with traces along two separating curves (partial products) and two additional triple-trace parameters measuring the corresponding complex twists; for n = 6 one analogously takes three separating curves and three twist parameters. The key point for what follows is not the uniqueness of such a choice but the fact that provides a closed symbolic calculus: once a generating set is fixed, the mapping class action becomes a list of explicit rational formulas in these trace generators.
Trace coordinates are intrinsic but tend to obscure the combinatorics
of mapping classes. To obtain simple birational formulas we pass to a
cluster-type atlas on a Zariski-open set, as follows. Let U ⊂ Y(0, n, 2)irr
be the locus where each local monodromy Ai is
diagonalizable with two distinct eigenlines in ℙ1; on U we can choose at each puncture a
framing line ℓi ⊂ ℂ2
given by one of the eigenlines. (For our purposes the particular choice
is irrelevant: different choices are related by an elementary birational
symmetry, and all constructions descend to PGL2 where appropriate.) On U one can attach to any ideal
triangulation T of the n-punctured sphere a set of
Xe ∈ ℂ×, e ∈ E(T),
defined as explicit cross ratios of the four framed lines adjacent to
the edge e after transporting
them to a common fiber by parallel transport along the arcs of T. These Xe are regular
on U and provide a birational
chart U⇢(ℂ×)3n − 6.
The local monodromy (conjugacy class) parameters appear as : for each
puncture xi there is a
monomial in the adjacent Xe (product with
signed exponents determined by T) which depends only on the
conjugacy class at xi; thus, upon
restriction to Y(C), exactly n independent monomials become fixed
and the remaining 2n − 6
parameters serve as coordinates.
The utility of these coordinates is that a change of triangulation by
a single flip induces an explicit birational change of variables. Let
T′ be obtained from
T by flipping an edge k in a quadrilateral. Denote by
ε = (εij)
the signed adjacency matrix of the quiver associated to T (constructed by placing an
oriented 3-cycle in each triangle and summing contributions along
edges). Then the flip T ⤳ T′ acts by
the standard 𝒳-mutation rule
where sgn (m) ∈ {−1, 0, 1}.
Formula is valid in any seed; for n = 5, 6 all εik
belong to {0, ±1, ±2}, so the exponents
are small and the resulting maps are easily iterated.
We will fix once and for all a reference triangulation Tn for each n ∈ {5, 6}. The corresponding seed provides 3n − 6 rational functions on U, and, after freezing the n puncture Casimirs, yields 2n − 6 mutable coordinates on each stratum Y(C) ∩ U. All subsequent computations of PMod0, n-dynamics in coordinates will be performed in this atlas and its images under flips.
The PMod0, n-action on tuples
(A1, …, An)
is induced by Hurwitz moves. Since the action on trace functions is
defined by substitution (Ai) ↦ (Ai′)
followed by reduction using , every braid generator σi yields an
explicit rational self-map of Y(0, n, 2) in any fixed
trace generating set. In particular, on the level of the tuple we
have
σi: (Ai, Ai + 1) ↦ (Ai + 1, Ai + 1−1AiAi + 1),
so that the transformed trace generators are obtained by taking traces
of words in the new Aj′
and then rewriting them back in the chosen trace coordinates. This gives
a fully explicit (and exact) description of the action on ℂ[Y(0, n, 2)], albeit one
whose closed form depends on the chosen generating family.
In cluster coordinates, the same mapping classes become substantially simpler. A basic fact is that a braid move exchanging two punctures corresponds, at the level of ideal triangulations, to a finite sequence of flips; thus its action on the X-coordinates is the composition of the corresponding mutations , followed by a relabeling of punctures if we pass from pure to full mapping class group. For PMod0, n we keep punctures fixed, so only mutations occur.
For concreteness we choose a finite generating set of PMod0, n consisting of
pure braids that drag one puncture around another. On the braid group
side these may be represented by the standard pure braid words
Pi, j := σj − 1⋯σi + 1 σi2 σi + 1−1⋯σj − 1−1 (1 ≤ i < j ≤ n),
whose images generate the pure mapping class group of the n-punctured sphere. Each Pi, j
is realized on the triangulation side by a loop in the flip graph of
triangulations; therefore, on any cluster chart it acts by a birational
automorphism given by an explicit word in mutations μk (and no
permutations). In practice, for n = 5 and n = 6 we precompute such mutation
words for a small generating subset of the Pi, j,
and we record the resulting birational maps as explicit rational
transformations of the 2n − 6
mutable variables (with the n
Casimirs fixed).
Two features of these formulas will be used repeatedly later. First, every generator acts by Laurent transformations: each new coordinate is a Laurent polynomial in the old ones with coefficients in ℤ and with explicit dependence on the frozen puncture parameters. Second, the Jacobian of a mutation has determinant ±1 on the logarithmic torus (ℂ×)2n − 6, so fixed-point computations can be carried out by writing down the equations Xe′ = Xe for the relevant generators and eliminating variables by elementary algebra. This is the point at which the dichotomy of the next section becomes visible: on one hand, positive-dimensional fixed loci correspond to underdetermined systems in these coordinates; on the other hand, isolated finite-orbit points are detected by the presence of enough independent fixed-point equations to cut the local dimension to zero.
In this section we formalize the elementary geometric dichotomy
underlying Theorem~A: a finite PMod0, n-orbit either
arises from a positive-dimensional fixed locus (the
pullback regime''), or else it is cut out as an isolated fixed point of a finite-index stabilizer (theKatz–reflection
regime’’). We also record a practical criterion, phrased in local
coordinates, for detecting when a finite-orbit point is isolated.
Let Y := Y(0, n, 2)
and let Yirr ⊂ Y denote
the irreducible locus. For a subgroup Γ ⊂ PMod0, n we
write
YΓ := {y ∈ Y: γ ⋅ y = y for
all γ ∈ Γ},
the (scheme-theoretic) fixed locus of Γ. If [ρ] ∈ Y has finite PMod0, n-orbit, then its
stabilizer
Stab([ρ]) := {γ ∈ PMod0, n: γ ⋅ [ρ] = [ρ]}
has finite index, since |PMod0, n ⋅ [ρ]| = [PMod0, n : Stab([ρ])].
Thus finite orbit points are precisely those which are fixed by some
finite-index subgroup. What matters for us is the geometry of the
corresponding fixed locus at [ρ].
The definition is insensitive to replacing Γ by a finite-index subgroup Γ′ ⊂ Γ: if Γ′ has finite index in Γ, then YΓ ⊂ YΓ′ and the local dimension at [ρ] is the same, because both fixed loci are cut out by finitely many algebraic equations in any affine chart around [ρ].
The following reformulation separates the classification problem into two regimes.
In case~(2), the point [ρ] lies on a positive-dimensional locus on which Γ acts trivially. Later, via Corlette–Simpson and Loray–Pereira–Touzet, this geometric non-isolation is converted into orbifold factorization of ℙρ, hence into the pullback-type description.
To decide whether we are in case~(1) or (2), it is useful to express
dim[ρ]YΓ
in terms of an invariant subspace in the Zariski tangent space. On the
irreducible locus the character variety is well behaved, and the
standard deformation theory applies: if 𝔤 = 𝔰𝔩2(ℂ) and Ad ρ denotes the adjoint local
system, then (at smooth points) the tangent space satisfies
T[ρ]Y ≅ H1(X, Ad ρ),
and similarly, on a fixed local monodromy stratum Y(C) one replaces
H1 by the
appropriate parabolic cohomology. The group PMod0, n acts on Y through outer automorphisms of
π1(X),
hence induces a linear action on T[ρ]Y.
The following is the usual fixed-point criterion.
Two remarks are important in practice. First, smoothness at [ρ] is automatic on a Zariski-open subset of Yirr; when [ρ] lies on a singular stratum, the equality of dimensions can fail, but the implication ``(T[ρ]Y)Γ ≠ 0 ⇒ dim[ρ]YΓ > 0’’ remains a reliable sufficient criterion for non-isolation. Second, in our concrete computations we do not explicitly build H1(X, Ad ρ); instead we implement the same test using Jacobians of the birational coordinate transformations from 3.
Fix a local monodromy stratum Y(C) and a cluster
chart on U ∩ Y(C)
as in 3.2, so that we have mutable coordinates
X = (X1, …, X2n − 6) ∈ (ℂ×)2n − 6
after freezing the n puncture
Casimirs. Any mapping class γ ∈ PMod0, n
acts by a birational transformation
Fγ: (ℂ×)2n − 6⇢(ℂ×)2n − 6, X ↦ X′.
Now let [ρ] ∈ U ∩ Y(C)irr
have finite orbit and let Γ ⊂ PMod0, n be
a finite-index subgroup fixing it. Choose a finite generating set γ1, …, γm
of Γ (in computations we
obtain such a set from explicit stabilizing braid words). Consider the
fixed-point equations in the cluster chart:
These equations cut out the intersection U ∩ Y(C) ∩ YΓ
inside the torus chart. The local dimension at X(ρ) can be read
off from the Jacobian of the defining map
Φ(X) := (Fγ1(X) − X, …, Fγm(X) − X),
after choosing local coordinates (e.g. logarithmic coordinates xi = log Xi
on a simply connected neighborhood).
In computations we typically apply Criterion~ in one of two equivalent forms.
Working in exact arithmetic over a number field containing the coordinates of [ρ], we evaluate the Jacobian matrices DFγk at X(ρ). If the block matrix obtained by stacking DFγk − I has rank < 2n − 6, then the intersection of kernels is nonzero and [ρ] is certified non-isolated (as soon as [ρ] lies in the chart U and is a fixed point of the chosen Γ).
We form the ideal generated by the equations in the coordinate ring of the torus chart (with frozen Casimirs specialized to the prescribed local monodromy data). A Gröbner basis computation (again in exact arithmetic) yields the Krull dimension of the corresponding affine scheme; if it is > 0, then [ρ] lies on a positive-dimensional fixed locus. This test is slower but robust near singular points and serves as a cross-check for the linear test.
The conceptual role of these tests is the following. If a finite-orbit point is non-isolated, then there is room to move while staying fixed under a finite-index subgroup; in the cluster chart this manifests as underdetermined fixed-point equations, or equivalently as a nontrivial invariant tangent direction. Conversely, when the fixed-point equations cut out a zero-dimensional germ, we are in the isolated regime, where one expects rigidity phenomena strong enough to force bounded Katz reduction.
We will use the fixed locus alternative (Lemma~) and the practical non-isolation criteria above as the entry point for the two classification arguments: in the next section we treat the positive-dimensional regime via orbifold pullbacks, and later we treat the isolated regime via Katz operations and finite complex reflection groups.
We treat the non-isolated regime from Lemma~. Throughout we fix n ∈ {5, 6} and a finite PMod0, n-orbit [ρ] ∈ Y(0, n, 2)irr which is isolated, so that [ρ] lies on a positive-dimensional fixed locus YΓ for some finite-index Γ ⊂ PMod0, n. The goal of this section is to explain how Corlette–Simpson and Loray–Pereira–Touzet convert this geometric hypothesis into an explicit pullback description, and how for n = 5, 6 the resulting pullback data form a finite effectively enumerable list 𝒫n.
Let ℙρ : π1(X) → PGL2(ℂ)
be the associated projective representation. By Lemma~2
(Corlette–Simpson/LPT pullback criterion in this setting), the existence
of a positive-dimensional Γ-fixed locus through [ρ] forces ℙρ to factor through an orbifold
curve. Concretely, after replacing the target orbifold by its
normalization and using that X ⊂ ℙ1 is a punctured
sphere, we may assume that there exists a rational map
g : ℙ1 → ℙ1
and a projective local system W on ℙ1 \ {0, 1, ∞} such that
where g|X : X → ℙ1 \ {0, 1, ∞}
is understood as a morphism of punctured spheres after possibly
enlarging the puncture set of X by removing any point where g fails to be a covering map. In
particular, the set of punctures {x1, …, xn}
consists of (i) points mapping to {0, 1, ∞}, and (ii) additional points above
which g is ramified but whose
images are not in {0, 1, ∞}; these
latter punctures are precisely the ``auxiliary points’’ in the informal
formulation of pullback type.
The passage from to an SL2-local system requires a lift from PGL2 to SL2. Since we work up to finite-order rank-one twists, we may (and do) absorb the obstruction into a character χ : π1(X) → ℂ× of finite order; equivalently, we consider ρ ⊗ χ−1 when convenient. Thus the pullback regime is naturally formulated projectively, while the rank-one twist records the difference between PGL2 and SL2 data.
Fix local monodromy conjugacy classes for W at 0, 1, ∞, and denote by M0, M1, M∞ ∈ PGL2(ℂ)
the corresponding local monodromies, with M0M1M∞ = 1.
The orbifold structure associated to W is encoded by the orders
p := ord(M0), q := ord(M1), r := ord(M∞),
allowing p, q, r = ∞ when
the corresponding element has infinite order. If y ∈ ℙ1 satisfies g(y) = 0 and ey is the
ramification index of g at
y, then the projective local
monodromy of g*W around a
small loop about y is
conjugate to M0ey;
similarly for points above 1 and ∞. Therefore, to realize a prescribed local
monodromy stratum on X, the
ramification indices above {0, 1, ∞}
must be compatible with the desired conjugacy classes of ℙρ(γi).
In the simplest (and most frequent) situation for finite-orbit
phenomena, the compatibility takes the form of a divisibility
constraint: if we want the pullback to have trivial projective local
monodromy at a ramified point y, then we require p ∣ ey
(and similarly q ∣ ey,
r ∣ ey
above the other marked values). More generally, if we require that ℙρ(γi)
have order di, then di must divide
p/gcd (p, ey)
in the finite order case, with the evident modifications when p = ∞.
Auxiliary punctures arise from points y where g is ramified but g(y) ∉ {0, 1, ∞}. Such a point does not introduce a singularity for W itself, but it does obstruct the restriction of g to a genuine topological covering map. Removing these points from the domain yields a map of punctured spheres g|X : X → ℙ1 \ {0, 1, ∞} inducing the pullback on π1. In the pullback classification problem, these points contribute punctures of X whose projective local monodromy is necessarily trivial (since W is nonsingular at the image), and hence their SL2-local monodromy is of the form ±I up to twisting.
We encode the covering-theoretic part of the pullback construction by a branched-cover passport.
Two passports are considered equivalent if they differ by post-composition of g with an automorphism of ℙ1 preserving {0, 1, ∞}, by pre-composition with an automorphism of ℙ1 (which reparametrizes the domain punctures), and by Hurwitz equivalence on the branch cycles. This matches the equivalences used later in the global classification: permutation of punctures, braid/Hurwitz moves, and reparametrization.
For n = 5, 6 there are only
finitely many such passports. The essential reason is the
Riemann–Hurwitz formula together with the constraint that |S| = n is small. If g has degree d and ramification indices {ey}, then
−2 = d(−2) + ∑y ∈ ℙ1(ey − 1), i.e. ∑y(ey − 1) = 2d − 2.
Since every point of ramification belongs to S, and |S| = n, the total
ramification 2d − 2 is
supported on at most n points.
This forces d to be bounded in
terms of n (indeed, one cannot
have large d while keeping the
ramification supported on only 5 or
6 points). Incorporating the further
condition that g−1({0, 1, ∞}) ⊂ S
refines the bound and yields a finite search space for the partitions
above 0, 1, ∞. This is the content of
Lemma~3, which we use as a finiteness-and-effectivity input.
Given a passport, we construct a candidate finite-orbit point as follows. Choose a projective local system W on ℙ1 \ {0, 1, ∞} with prescribed local monodromy orders (p, q, r); in practice we take W to be a projectivization of a hypergeometric SL2-local system when such a lift exists, or otherwise we work projectively and later choose a finite-order twist to lift. Form the pullback (g|X)*W on X = ℙ1 \ S, and then select an SL2-lift ρ of ℙρ by fixing determinants (equivalently, by choosing a finite-order rank-one twist). The resulting [ρ] ∈ Y(0, n, 2) belongs to a positive-dimensional fixed locus for a finite-index subgroup coming from the deck transformations and from the symmetry of the base three-punctured sphere, and hence it is a non-isolated finite-orbit point in the sense of Definition~.
The classification statement for pullback type is then organized by the finite set 𝒫n of equivalence classes of passports together with a finite set of base projective monodromy types W (equivalently, conjugacy classes of triples (M0, M1, M∞) in PGL2 satisfying the orbifold constraints). We emphasize that 𝒫n is finite and effectively enumerable: the passport enumeration is finite by Lemma~3, and for each passport only finitely many base monodromy types occur in the finite-orbit setting considered here (as in the constructions appearing in the work of Diarra, Doran, and Kitaev, and in the geometric framework of Corlette–Simpson/LPT). In particular, for n = 5, 6 the pullback side contributes no genuinely new infinite families beyond those arising from varying the positions of punctures within a fixed passport class; at the level of PMod0, n-orbits, this is reflected by the presence of a positive-dimensional Γ-fixed locus.
We record the outcome in a form parallel to the Katz–reflection side handled later.
The proof is a direct assembly of Lemma~2 (orbifold factorization), the reduction to maps g : ℙ1 → ℙ1 with target ℙ1 \ {0, 1, ∞}, and Lemma~3 (finiteness of passports), together with a case-by-case identification of the base projective monodromy types allowed by the finite-orbit hypothesis. We postpone the explicit presentation of 𝒫n and representative maps g to the appendices, where each passport is recorded either by branch cycles (σ0, σ1, σ∞) or by an explicit rational function g over a number field.
This completes the treatment of the positive-dimensional fixed-locus regime. In the remainder of the paper we therefore concentrate on isolated finite-orbit points; the next section introduces the Katz operations (rank-one twists and middle convolution) which provide the main mechanism for reducing isolated points to finite complex reflection group data.
In the isolated regime, our main tool is the package of : finite-order rank-one twists and (multiplicative) middle convolution MCλ. We recall these operations in the form convenient for explicit manipulation of tuples (A1, …, An) ∈ SL2(ℂ)n with ∏iAi = I, and we record the basic properties needed later: control of local monodromy, control of rank, and equivariance with respect to Hurwitz moves (hence preservation of finiteness of mapping class orbits).
Let χ : π1(X) → ℂ×
be a rank-one character. Given a representation ρ : π1(X) → GL(V),
we define the twist ρ ⊗ χ by
(ρ ⊗ χ)(γ) := ρ(γ) χ(γ) (γ ∈ π1(X)).
On tuples Ai = ρ(γi)
this is simply
$$
A_i\longmapsto A_i':=t_iA_i,\qquad
t_i:=\chi(\gamma_i)\in\mathbb{C}^\times,
\qquad \prod_{i=1}^n t_i=1,
$$
so that ∏iAi′ = I
still holds. In the SL2
setting, a twist does not necessarily preserve determinant 1 unless χ2 is trivial; however,
since we only twist by characters of finite order and we work up to
simultaneous conjugacy in SL2, this mismatch is harmless:
twisting is an equivalence operation in our classification scheme, and
it is used precisely to normalize eigenvalue patterns before and after
middle convolution.
Two features will be used repeatedly. First, twisting changes local eigenvalues by scalars but preserves Jordan form sizes; in particular, the dimensions dim ker (Ai − I) are unchanged unless ti moves an eigenvalue across 1. Second, twisting is PMod0, n-equivariant (indeed natural in π1(X)), so it carries finite orbits to finite orbits.
We recall the multiplicative middle convolution MCλ for λ ∈ ℂ× \ {1} in the concrete form of Dettweiler–Reiter . We describe it on tuples, since this is the form used in certified computations.
Let V be a complex vector
space of dimension r, and
let
A = (A1, …, An) ∈ GL(V)n, A1⋯An = I.
Set Ṽ := V⊕n
with coordinates (v1, …, vn).
For each i ∈ {1, …, n} define an
endomorphism B̃i ∈ GL(Ṽ)
by
where the correction term occupies the slots i, i + 1, …, n
and
w := v1 + ⋯ + vi − 1 + (Ai − I)vi.
One checks directly that B̃1⋯B̃n = I,
so $\widetilde{\mathbf{B}}=(\widetilde{B}_1,\dots,\widetilde{B}_n)$
defines a new representation of π1(X) on Ṽ; this is the (multiplicative) of
A with parameter λ.
Middle convolution is obtained by passing to a natural quotient that
removes the ``obvious’’ invariant subspaces created by the construction.
Define
K̃ := ⨁i = 1nker (Ai − I) ⊂ Ṽ, L̃ := {(v, …, v) ∈ Ṽ : v ∈ V}.
Then K̃ and L̃ are $\widetilde{\mathbf{B}}$-stable, and we
set
V′ := Ṽ/(K̃ + L̃), Bi := the
induced operator on V′.
The resulting tuple B = (B1, …, Bn) ∈ GL(V′)n
with ∏iBi = I
is, by definition,
MCλ(A) := B.
This construction is functorial for morphisms of representations and
depends algebraically on the entries of the Ai and on λ. In particular, if the Ai are defined
over a number field and λ is
algebraic, then so is MCλ(A),
and one can compute it in exact arithmetic by working with an explicit
basis of V′.
The first structural input is the rank change. Let r = dim V and mi := dim ker (Ai − I).
In our n-punctured-sphere
normalization we may regard ∞ as an
additional point with monodromy A∞ := (A1⋯An)−1 = I.
Since λ ≠ 1, we have ker (λ−1A∞ − I) = 0.
Dettweiler–Reiter show that
Thus MCλ typically
increases rank, but it can also decrease rank when sufficiently many
Ai have
nontrivial 1-eigenspace. In the rank
2 case (r = 2), becomes
$$
r' = 2(n-2) - \sum_{i=1}^n \dim\ker(A_i-I),
$$
which makes transparent why finite-order twists are useful: by moving
eigenvalues to 1 at selected punctures,
one can force MCλ
to land in ranks 3 (when n = 5) or 4 (when n = 6), which are precisely the
dimensions where the Shephard–Todd classification of finite complex
reflection groups is available in a manageable form.
We also record the eigenvalue-level effect on local monodromy, which
we use as a bookkeeping device. Under mild nonresonance hypotheses (in
particular, avoiding simultaneous appearance of eigenvalue 1 and the parameter λ in incompatible Jordan
configurations), the spectrum of the local monodromy transforms by the
rule
counted with multiplicity. In words: eigenvalues different from 1 are scaled by λ, while new 1-eigenspaces are created so as to match the
rank change. For r = 2 this
forces Bi
to be close to a pseudoreflection precisely when Ai has a large
1-eigenspace after twisting; this is
the local mechanism behind the pseudoreflection phenomenon exploited in
the reduction theorem of the next section.
Middle convolution is an equivalence operation on the appropriate
open subcategories: applying MCλ−1 after
MCλ recovers the
original representation up to a canonical rank-one twist, again under
standard nonresonance hypotheses. Concretely, there exists a character
δ (explicit in terms of
determinants of local monodromy) such that
Thus, in our classification, we may apply middle convolution and later
undo it without losing information, provided we keep track of
finite-order twists.
Equally important is compatibility with the Hurwitz (braid) action.
Recall that Bn acts on
tuples A = (A1, …, An)
by the standard Hurwitz moves, preserving the product relation. The
construction is natural with respect to automorphisms of π1(X), and in
particular it is compatible with the braid action in the sense that for
each braid β ∈ Bn
there is a canonical simultaneous conjugacy between the tuples
MCλ(β ⋅ A) and β ⋅ MCλ(A).
After passing from braids to mapping classes, this implies that orbit
finiteness is preserved by Katz operations:
We will apply Lemma~ in both directions: starting from an SL2 tuple with a finite orbit, we pass to a Katz-reduced tuple of higher rank where finiteness can be certified by finite group considerations; conversely, from a finite complex reflection group datum we produce an SL2 tuple and deduce finiteness of its mapping class orbit by equivariance.
For later certified enumeration we require a concrete representation of MCλ(A) as matrices over a number field. The Dettweiler–Reiter construction provides this directly.
Starting from A over a field K ⊂ ℂ and λ ∈ K×, we form the block operators B̃i on Ṽ = V⊕n ≅ Krn using . We then compute bases for K̃ and L̃, form a basis of the quotient V′ = Ṽ/(K̃ + L̃), and finally express each induced Bi in that basis. This yields an explicit tuple (B1, …, Bn) ∈ GLr′(K)n satisfying ∏iBi = I.
Two remarks are relevant for correctness. First, all steps can be performed in exact linear algebra over K, so no numerical approximation enters the proof. Second, irreducibility and local monodromy type of MCλ(A) can be checked by routine exact computations (invariant subspaces, eigenvalues and Jordan forms, and determinant constraints), and these checks form part of the certificates described later in Lemma~.
With these preliminaries in place, we can formulate and prove the bounded Katz reduction statement for isolated finite orbits, where the special features of n = 5, 6 enter through the rank formula and the resulting restriction to complex reflection groups in dimensions 3 and 4.
In this section we explain why, for n ∈ {5, 6}, an finite PMod0, n-orbit in rank 2 can be reduced by a bounded sequence of Katz operations to a finite monodromy representation in rank n − 2 whose local monodromies are pseudoreflections at n − 1 punctures. This is the mechanism underlying the ``Katz–reflection’’ branch of the dichotomy.
Throughout we fix n ∈ {5, 6} and an irreducible SL2-representation ρ on X = ℙ1 \ {x1, …, xn} with finite PMod0, n-orbit, and we assume that [ρ] is isolated as a finite orbit point in the sense of the Global Context. By Lemma~, we are free to replace ρ by a finite-order twist and by any Katz transform MCλ (whenever defined), without affecting finiteness of the orbit; isolation will be used to show that suitable choices exist and can be made in boundedly many steps.
We isolate the reduction statement in a form tailored for the subsequent enumeration step.
The proof is a combination of (i) a local eigenvalue analysis using and , (ii) a rigidity consequence of isolation, and (iii) a finiteness criterion for pseudoreflection-generated monodromy in ranks 3 and 4.
We briefly explain why n ∈ {5, 6} is special. Starting from
rank 2, the rank formula gives
$$
\dim\mathrm{MC}_{\lambda^{-1}}(\rho)\ =\ 2(n-2)\ -\ \sum_{i=1}^n m_i,
\qquad m_i:=\dim\ker(A_i-I),\ \ A_i:=\rho(\gamma_i).
$$
Thus, to land in rank n − 2 we
must have
Since mi ∈ {0, 1, 2}
in rank 2, the constraint is very
restrictive for n = 5, 6: it
forces only finitely many combinatorial patterns for the 1-eigenspace dimensions. This finiteness is
the first boundedness input: up to twisting and reordering punctures
(within PMod0, n),
we can enforce a small list of candidate patterns for (m1, …, mn)
compatible with irreducibility and isolation.
The second n = 5, 6 feature is the target rank itself: n − 2 ∈ {3, 4}. In these dimensions, the Shephard–Todd classification controls irreducible finite complex reflection groups, so once we know that ρ̂ has finite image and is generated by pseudoreflections, the ambient group is forced into a finite explicit list.
We now extract from a convenient criterion for pseudoreflection monodromy at a puncture. Write r = 2 and r′ = n − 2, and let Ai have eigenvalues αi, αi−1. Ignoring the standard nonresonance exclusions (which we enforce by a preliminary twist and, if needed, a change of representative in the PMod0, n-orbit), implies that the multiplicity of the eigenvalue 1 in Bi equalsThus the desired pseudoreflection behavior can be arranged either by creating 1-eigenspaces (via finite-order twists moving an eigenvalue to 1) or by matching one eigenvalue to λ (via choosing λ appropriately, again up to finite-order twisting). The point of is that it decouples the pseudoreflection requirement from the rank constraint : we can have many pseudoreflections even when ∑mi is small, by using the si-contribution.
For n = 5, 6 we may combine
with as follows. To produce pseudoreflections at n − 1 punctures, it suffices to
arrange that holds for i = 1, …, n − 1, while
Bn is left
unrestricted. One compatible pattern is:
mi = 1 for
exactly n − 2
indices, mj = 0 for the remaining two
indices,
and among the two indices with m = 0, we enforce s = 1 at exactly one of them (so
that one of those punctures still yields a pseudoreflection). This is
precisely the configuration that produces n − 1 pseudoreflections while
satisfying ∑mi = n − 2.
We indicate where isolation enters. If [ρ] is isolated as a finite orbit point, then, by definition, there is no positive-dimensional family of representations fixed by the stabilizer of [ρ]. In practice, this implies that the local conjugacy classes (C1, …, Cn) of ρ cannot vary continuously while preserving the finite-orbit constraint, and hence the local eigenvalue data must lie in a discrete subset of (ℂ×)n determined by the periodicity constraints coming from the mapping class action.
Concretely, we use isolation in two places.We emphasize that we do not claim every local eigenvalue of ρ is torsion ; rather, we claim that after boundedly many Katz normalizations one reaches a representation ρ̂ in rank n − 2 for which the eigenvalue of each pseudoreflection Bi is λ, hence of finite order.
Assuming we have produced ρ̂ of rank n − 2 with B1, …, Bn − 1 pseudoreflections and λ a root of unity, we conclude finiteness of the monodromy group as follows. Let Ĝ be the subgroup of GLn − 2(ℂ) generated by the image of ρ̂, equivalently by B1, …, Bn. Since B1, …, Bn − 1 are pseudoreflections with nontrivial eigenvalue λ, all of them have finite order dividing ord(λ). Moreover, the tuple (B1, …, Bn) has finite PMod0, n-orbit by Lemma~. In ranks 3 and 4, this combination (finite orbit together with pseudoreflection local monodromy at n − 1 punctures) forces Ĝ to be finite: if Ĝ were infinite, then the corresponding point of the relevant character variety would lie on a positive-dimensional family invariant under a finite-index stabilizer, contradicting the isolated nature of the original orbit once Katz invertibility is taken into account. We record this implication as a lemma, whose proof is a rigidity-by-contradiction argument exploiting the low-dimensional constraint n − 2 ∈ {3, 4}.
Putting together the local pseudoreflection criterion , the rank constraint , and Lemma~, we obtain Proposition~. The constant Ln arises because each step (choice of λ, placement of the mi = 1 conditions via twists, and avoidance of resonant loci) comes from a finite menu of possibilities when n = 5, 6, and invertibility prevents unbounded wandering in the Katz graph.
This completes the reduction to finite complex reflection monodromy in dimension n − 2. In the next section we exploit this reduction by filtering the Shephard–Todd list in dimensions 3 and 4 and then pushing the resulting reflection data back to rank 2 via explicit Katz operations.
We explain how Proposition~ turns the classification of isolated finite orbits into a finite enumeration problem in dimensions 3 and 4, and how we produce the explicit lists ℒ5 and ℒ6 of Katz–reflection data. The outcome of the present section is a finite collection of candidates, each equipped with enough explicit structure to be pushed forward to rank 2 by Katz operations and then subjected to orbit and irreducibility verification.
Fix n ∈ {5, 6} and set d := n − 2 ∈ {3, 4}. By Proposition~, any isolated finite PMod0, n-orbit in rank 2 is related, by a bounded chain of invertible Katz operations, to a rank d representation ρ̂ with finite image generated by a tuple (B1, …, Bn) in which B1, …, Bn − 1 are pseudoreflections. In particular the image $\widehat{G}:=\widehat{\rho}(\pi_1(X))\subset \GL_d(\C)$ is an irreducible finite complex reflection group. By the Shephard–Todd classification, there are only finitely many such Ĝ in each of the dimensions d = 3, 4.
Thus the enumeration task becomes: for each irreducible Shephard–Todd group $G\subset \GL_d(\C)$, list (up to the standard equivalences) the tuples of pseudoreflections (B1, …, Bn − 1) in G that can occur as local monodromies after Katz reduction, together with the corresponding parameter λ and the induced class of Bn = (B1⋯Bn − 1)−1. We package this information as a Katz–reflection datum (G, λ, D) in the sense of the Global Context, where D = (D1, …, Dn − 1) records the conjugacy classes of the pseudoreflections Bi.
Two constraints are imposed at the outset.
After these two constraints, only finitely many (G, λ) remain, and for each such pair only finitely many relevant pseudoreflection conjugacy classes exist. Lemma~5 of the Global Context is precisely the abstract finiteness statement behind this filtering.
For each admissible (G, λ), we construct ordered tuples (B1, …, Bn − 1) with Bi ∈ Di for selected conjugacy classes Di of pseudoreflections of determinant λ. We then set Bn := (B1⋯Bn − 1)−1 so that $\prod_{i=1}^n B_i=I$, as required for a representation of π1(X).
Two equivalence relations must be accounted for already at the level of tuples in G.
Accordingly, we proceed as follows. For each ordered choice of conjugacy classes D = (D1, …, Dn − 1) of pseudoreflections with determinant λ, we search for tuples B1 ∈ D1, …, Bn − 1 ∈ Dn − 1 such that the resulting Bn makes ⟨B1, …, Bn⟩ equal to G (or at least an irreducible reflection subgroup satisfying the same Katz constraints). We then quotient the resulting set by the combined action of simultaneous conjugacy and Hurwitz moves. The output of this step is a finite list of braid orbits of tuples in G, each labeled by (G, λ, D) together with the conjugacy class of Bn determined by the product relation.
A minor subtlety is that the ordering of punctures is part of the input to the pure mapping class group action. We therefore keep the tuples ordered at this stage; any later identification arising from permuting punctures is handled only when passing from PMod0, n to Mod0, n or when implementing the ``natural equivalences’’ in the final classification statement.
Given a tuple (B1, …, Bn) in $G\subset\GL_d(\C)$ produced above, we regard it as the monodromy tuple of a rank d local system ρ̂ on X. The goal is to construct from ρ̂ a rank 2 local system ρ by applying inverse Katz operations, in a way consistent with Proposition~. Concretely, we apply a finite sequence of middle convolutions $\MC_{\mu}$ and rank-one twists (and inverses) so that the resulting representation has rank 2 and lands in $\SL_2(\C)$.
We emphasize three points.
The output of this step is, for each admissible (G, λ, D), an explicit rank 2 monodromy tuple $(A_1,\dots,A_n)\in \SL_2(\overline{\Q})^n$ with ∏iAi = I, together with a recorded Katz chain witnessing that it arises from the reflection tuple. We include (G, λ, D) in ℒn precisely when at least one such chain exists and yields an irreducible rank 2 representation.
Although finiteness of the PMod0, n-orbit is formally preserved under invertible Katz operations, the enumeration produces many spurious candidates: some yield reducible rank 2 representations, some land outside the domain of invertible middle convolution, and some coincide with others after applying the natural equivalences. We therefore attach to each surviving candidate a verification package suitable for later certification.
At the level of rank 2 tuples (A1, …, An), we perform the following checks.
The practical effect is that each element of ℒn is not merely an abstract reflection datum, but a datum equipped with at least one explicit rank 2 representative and with enough orbit information to be checked independently. The next section explains how these verification targets are turned into proof-grade certificates and how completeness (every isolated finite orbit arises from the enumeration) is deduced.
In this section we justify the two formal claims that are implicit in the construction of the lists ℒn and 𝒫n, namely: (i) , i.e. every finite PMod0, n-orbit in Y(0, n, 2)irr is represented by (at least) one element of ℒn ∪ 𝒫n, and (ii) , i.e. no orbit type is counted twice once we quotient by the standard relations (Hurwitz moves, simultaneous conjugacy, permutation of punctures when relevant, and invertible Katz operations). We also isolate the points at which certified computation enters and explain how the resulting output can be treated as proof data rather than as heuristic evidence.
We restrict attention to isolated finite orbit points; the non-isolated case is reduced to pullback type by Lemma~2 in the Global Context and is treated by the passport enumeration of Lemma~3. Thus we assume throughout that [ρ] ∈ Y(0, n, 2)irr has finite PMod0, n-orbit and is isolated in the sense of the Global Context.
The starting point is Proposition~, which we use as a black box reduction theorem. Applied to [ρ], it produces:such that $\widehat{\rho}\cong
\MC_{\lambda^{-1}}(\rho\otimes\chi^{-1})$ (up to the recorded
chain), and such that the local monodromies of ρ̂ at n − 1 punctures are
pseudoreflections with common nontrivial eigenvalue λ (Lemma~4 in the Global Context).
Writing ρ̂ as a monodromy tuple
$(B_1,\dots,B_n)\in\GL_d(\C)^n$ with
∏iBi = I,
we obtain a finite irreducible complex reflection group
$$
G:=\langle B_1,\dots,B_n\rangle\subset \GL_d(\C),
$$
generated by pseudoreflections and acting irreducibly on $\C^d$.
By Shephard–Todd, the ambient possibilities for G are finite in each d ∈ {3, 4}. Our enumeration in
begins precisely by iterating over this finite Shephard–Todd list and,
for each group G, iterating
over pseudoreflection conjugacy classes of a fixed determinant λ. The key point for completeness is
that the reduction theorem forces λ across the n − 1 pseudoreflections.
Consequently, for the specific (G, λ) associated to [ρ], the ordered class vector
D = (D1, …, Dn − 1), Di := conjugacy
class of Bi in G,
is among the finite set of class-vectors inspected by the filtering
procedure.
Next we must ensure that our tuple search over (G, λ, D) actually finds a representative in the simultaneous conjugacy and Hurwitz class of (B1, …, Bn). This is formal: the tuple (B1, …, Bn) is itself a witness of existence for the constraints imposed in , namely product I, pseudoreflection condition at n − 1 punctures with eigenvalue λ, and irreducible generation of G. Thus the search space, which is finite and exhaustive for each fixed (G, λ, D), contains at least one element in the orbit of (B1, …, Bn) under the combined action of simultaneous conjugacy in G and Hurwitz moves. In particular, the reduced datum attached to [ρ] appears among the enumerated Katz–reflection data candidates.
Finally, we must pass from ρ̂ back to a rank 2 local system. Here boundedness is essential: Proposition~ provides a Katz chain of length ≤ Ln producing ρ from ρ̂ (up to a twist and the choice of λ). In our construction of ℒn we explore all admissible Katz graphs of length ≤ Ln starting from each reflection tuple, discarding branches that hit the resonant exceptional locus or fail rank-positivity. Since 𝒦 is one such admissible chain, it follows that our exploration produces at least one chain taking the enumerated reflection tuple back to a rank 2 tuple in $\SL_2$, and hence produces a representative of [ρ]. This gives the desired completeness statement for isolated finite orbits.
Summarizing the preceding discussion, we record the conclusion in the form used later for the global classification.
Completeness alone does not prevent multiple elements of ℒn ∪ 𝒫n from producing the same orbit type. We therefore impose and verify a non-redundancy condition: whenever two entries yield equivalent rank 2 points in Y(0, n, 2), they are already identified by the ``natural equivalences’’ built into the statement of Theorems~B and~C.
On the Katz–reflection side, there are two sources of redundancy.To address (1), we work with reflection data already quotiented by simultaneous conjugacy in G and Hurwitz moves. Thus, at the reflection level, we record only one representative per G-conjugacy class of Hurwitz orbit, and this removes the most immediate duplication.
To address (2), we use orbit-level invariants and explicit orbit computation. Concretely, for each rank 2 candidate tuple (A1, …, An) output by the Katz push-forward step, we compute its orbit under a fixed finite generating set of PMod0, n (or a fixed finite-index subgroup realized by explicit Hurwitz-type moves), obtaining a finite set S as in Lemma~6 of the Global Context. We then canonically normalize the tuples in S (for instance by choosing a preferred representative in each $\SL_2$-conjugacy class using trace coordinates on the character variety) and store the resulting normalized orbit as the orbit identifier of the candidate. Two candidates are declared equivalent if and only if these orbit identifiers coincide, possibly after applying the permitted permutation action of punctures in the Mod0, n setting.
Mathematically, the point is that the orbit identifier is not merely a computational heuristic: it is accompanied by a certificate that the computed set S is PMod0, n-stable and contains the original tuple. Hence equality of orbit identifiers implies equality of PMod0, n-orbits, and conversely, finiteness guarantees that a stable S is exactly the orbit. The remaining step is to check that whenever two entries of ℒn yield the same orbit identifier, they are related by the explicitly allowed equivalences. For Katz–reflection candidates this is done by exhibiting, as part of the output, an explicit chain of equivalences: a braid word sending one tuple to the other (when they lie in the same orbit) and, if necessary, an explicit comparison of Katz chains showing that both representatives reduce to the same reflection datum up to the already-quotiented actions.
On the pullback side 𝒫n, non-redundancy is treated similarly but with a different set of invariants: we use the branching passport of the map g, the orbifold type of the base projective local system on $\P^1\setminus\{0,1,\infty\}$, and the induced local conjugacy classes on X. Two pullback entries are identified if these discrete invariants agree and the corresponding induced rank 2 orbits coincide (again certified by explicit orbit computation at the end of the construction). In practice, the discrete invariants eliminate almost all collisions, and the final orbit check removes the remaining accidental identifications.
The preceding arguments reduce completeness and uniqueness to two kinds of finite tasks: (i) enumerating candidate data in a finite search space (Shephard–Todd groups, pseudoreflection classes, bounded Katz graphs, and pullback passports), and (ii) verifying that each surviving candidate indeed produces a finite orbit point in Y(0, n, 2)irr and that distinct candidates correspond to distinct orbit types modulo equivalence. Both tasks are implemented by computer algebra, but we insist on a proof-theoretic separation: the proof uses only statements that can be checked exactly from finite certificates.
Concretely, for each element of ℒn ∪ 𝒫n we store a certificate consisting of:Given such a package, every assertion needed later (finite orbit, membership in the irreducible locus, equivalence or inequivalence of candidates) reduces to finitely many exact computations in K, hence is mechanically checkable and independent of floating-point numerics or probabilistic tests.
From the logical perspective, the role of the computer is therefore confined to producing these certificates and to confirming the exhaustion of the finite search spaces that arise from Proposition~ and Lemmas~3–5. Once the certificates are fixed, the remainder of the argument is formal: Proposition~ supplies existence of a reduction into the enumerated reflection regime for isolated points; Lemma~2 and the passport finiteness supply the pullback regime for non-isolated points; and Lemma~6 supplies orbit finiteness and orbit equality checks for the explicit representatives. This is the point at which Theorems~B and~C become fully effective: the lists ℒn ∪ 𝒫n are not only finite, but each entry can be validated independently by verifying its certificate. The next section records the resulting representatives in tables and illustrates the two types by worked examples.
We record in this section explicit representatives for each orbit type occurring in Theorems~B and~C, together with a uniform set of discrete invariants that allow the reader to (i) reconstruct the corresponding point of Y(0, n, 2) exactly over a number field, (ii) certify finiteness of the PMod0, n-orbit by a direct check as in Lemma~6, and (iii) compare our output with previously known sporadic constructions (notably the icosahedral example for n = 5) and with earlier computational classifications of finite braid or mapping class group dynamics.
All full tables are placed in the appendices, since the essential point for the main text is not the raw volume of data but the fact that every entry is accompanied by a verification certificate in the sense of the Global Context. Here we explain the organization of the tables and work out a small number of representative examples in detail.
For n ∈ {5, 6} an entry of
ℒn ∪ 𝒫n
ultimately yields a rank 2 tuple
$$
(A_1,\dots,A_n)\in \mathrm{SL}_2(\overline{\mathbb{Q}})^n,\qquad
\prod_{i=1}^n A_i = I,
$$
considered up to simultaneous conjugacy in SL2 and the PMod0, n-action. In the
appendices we provide, for each orbit type, one such tuple with
coefficients in an explicit number field K, presented either as explicit
2 × 2 matrices in SL2(K) or, when more
compact, by a set of trace coordinates that uniquely determines the
SL2-conjugacy class away
from the reducible and resonant loci. The trace encoding we use is
standard: we list the local traces
ai := tr(Ai) (1 ≤ i ≤ n),
together with a fixed collection of mixed traces tr(AiAj)
sufficient to recover the point on the relevant affine chart of the
character variety (for n = 5
we use a Fricke-type chart with three mixed traces; for n = 6 we use an analogous chart with
six mixed traces). When the tuple is given by traces, the certificate
includes a reconstruction step producing an explicit matrix tuple over
K that realizes those traces
and satisfies the product relation.
On the Katz–reflection side we additionally record the Shephard–Todd label of the reflection group G, the common nontrivial eigenvalue λ of the pseudoreflections on the Katz-reduced side, the pseudoreflection conjugacy classes used at the n − 1 punctures, and an explicit Katz chain (twists and middle convolutions) connecting the reflection tuple to the rank 2 tuple. On the pullback side we record the passport of the rational map g : ℙ1 → ℙ1 (degree, branch cycle structure above 0, 1, ∞ and auxiliary branch points), and the base projective local system on ℙ1 \ {0, 1, ∞} (given by its local projective orders or, equivalently, by hypergeometric parameters when applicable).
We illustrate the Katz–reflection mechanism on the classical sporadic example for n = 5 whose projective monodromy is icosahedral (cf. Tykhyy). In our organization this orbit appears as a Katz–reflection entry whose reduction lands in dimension d = n − 2 = 3, with reflection group G isomorphic (projectively) to W(H3); the certificate in the appendix begins with a tuple of pseudoreflections (B1, …, B5) ∈ GL3(K)5 with ∏iBi = I, B1, …, B4 pseudoreflections of common nontrivial eigenvalue λ, and ⟨B1, …, B5⟩ irreducible.
The rank 2 representative is
obtained from (Bi) by a
recorded chain 𝒦 of length ≤ L5. Rather than
reproduce (Bi) and 𝒦 in full, we isolate the resulting 2 × 2 tuple and its orbit computation. Let
ω = e2πi/5
and set K = ℚ(ω).
Consider the matrices
$$
S=\begin{pmatrix}\omega & 0\\ 0 &
\omega^{-1}\end{pmatrix},\qquad
T=\frac{1}{\sqrt{5}}\begin{pmatrix}\omega^2-\omega^{-2} &
\omega-\omega^{-1}\\ \omega-\omega^{-1} &
-(\omega^2-\omega^{-2})\end{pmatrix}\in \mathrm{SL}_2(K(\sqrt{5})).
$$
(One checks det (T) = 1 using
sin2(π/5) + sin2(2π/5) = 5/4,
and T2 = −I, so
T has projective order 2.) Define a 5-tuple by
A1 = T, A2 = S, A3 = T, A4 = S−1, A5 = (A1A2A3A4)−1.
Then $\prod_{i=1}^5 A_i=I$ by
construction, and the projective subgroup of PGL2(ℂ) generated by the images of
S and T is icosahedral. The appendix
certificate replaces this illustrative choice by the specific tuple
produced by our Katz chain (with the same projective type), but the
verification procedure is identical: we compute the PMod0, 5-orbit of (Ai) under a
fixed generating set, using exact arithmetic in K, and obtain a finite set S closed under the generators. The
closure tables provide the certificate demanded by Lemma~6, and
irreducibility is certified by checking that no common eigenline is
preserved by (say) A1 and A2.
The comparison with the literature is then straightforward. First, the orbit size and the multiset of local traces match the invariants recorded for the icosahedral sporadic orbit in existing computations. Second, our entry additionally records the Katz–reflection provenance: a specific (G, λ, D) and a specific chain 𝒦 exhibiting the orbit as a push-forward from a pseudoreflection tuple. This extra structure is not present in purely dynamical enumerations, and it is precisely what makes the completeness argument effective in .
We next illustrate pullback type. Fix a projective local system W on ℙ1 \ {0, 1, ∞} with prescribed
projective local orders (p, q, r) and
choose a rational map g : ℙ1 → ℙ1
whose branching is supported on {0, 1, ∞} together with one auxiliary point.
The passport fixes the cycle types of the branch permutations above
0, 1, ∞ and the auxiliary branch point.
For a concrete instance, the appendices include degree-d maps with passports of the
form
0: (2, 2, …), 1: (3, 1, …), ∞: (5, …), aux: (2, 1, …),
chosen so that above {0, 1, ∞} there
are exactly n = 5 points where
the pulled-back local system has nontrivial local monodromy after
accounting for the orbifold divisibility constraints. The resulting
projective local system g*W on X = ℙ1 \ {x1, …, x5}
admits an SL2-lift V after a finite-order twist, and
V has finite PMod0, 5-orbit because the
corresponding point lies on a positive-dimensional fixed locus for a
finite-index subgroup of PMod0, 5 (equivalently, the
isomonodromy deformation is algebraic along the Hurwitz space determined
by the passport).
In the tables, such an entry is encoded by the passport and the base orbifold type (p, q, r), together with one explicit lifted SL2-tuple (Ai) and its finite orbit certificate. This makes the comparison between two pullback constructions transparent: two entries coincide if and only if their discrete pullback invariants agree and the final orbit identifiers agree. In particular, the ``family’’ aspect of pullback type is visible in the structure of the certificate: the stabilizer contains a subgroup arising from the deck transformations of the cover and from the braid relations dictated by the passport.
For n = 5, several sporadic finite orbits had been isolated previously by direct braid dynamics on trace coordinates; the icosahedral orbit is the most prominent. Our list ℒ5 ∪ 𝒫5 contains these sporadic examples, but reorganizes them into two conceptually rigid sources: pullback passports and Katz–reflection data. In particular, what appears as an isolated ``sporadic’’ orbit dynamically is, in our framework, the image of a pseudoreflection tuple in a finite complex reflection group under an explicitly bounded Katz chain. This explains, at the structural level, why such examples exist and why they are finite in number once n is fixed.
For n = 6, the novelty is more pronounced: allowing local monodromies of arbitrary order enlarges the landscape, and one sees reflection-type examples whose rank 2 local monodromies need not all be of finite order even though the Katz-reduced monodromy is finite. The tables make this phenomenon visible by listing, alongside the reflection group G ⊂ GL4, both the reduced pseudoreflection data and the resulting SL2 local traces; in several entries some ai = tr(Ai) are not of the form ζ + ζ−1 with ζ a root of unity. Such entries are absent from classifications that impose finite local monodromy a priori, and their existence is one of the reasons the bounded reduction statement (Proposition~D) is logically indispensable.
Finally, we emphasize that our tables are not intended as a mere compilation. Each row is a fully checkable mathematical object: from the stored matrices one verifies irreducibility, the product relation, and closure of the orbit under mapping class generators in exact arithmetic. Thus the comparison with earlier computations can be carried out at the level of certified equality of orbit identifiers, rather than at the level of matching numerical invariants. This is the sense in which the tables serve as explicit representatives of orbit types, rather than as suggestive numerical data.
For n ≥ 4, the character variety Y(0, n, 2) may be identified (on suitable open charts) with the Betti moduli of logarithmic SL2-connections on ℙ1 with poles at {x1, …, xn}, and the PMod0, n-action corresponds to analytic continuation of the isomonodromy connection along loops in ℳ0, n. In particular, a point [ρ] ∈ Y(0, n, 2) with finite PMod0, n-orbit yields an algebraic leaf of the isomonodromy foliation on the corresponding de Rham moduli space, hence an algebraic solution of the associated Schlesinger system. In the rank 2, genus 0 case, this is the classical mechanism by which finite mapping class dynamics produce algebraic solutions of Painlev'e~VI (for n = 4) and of Garnier systems (for n ≥ 5).
When n = 5, the isomonodromy system is the two-variable Garnier system, and when n = 6 it is the three-variable Garnier system. Theorems~B and~C therefore give a complete list, up to the standard equivalences, of algebraic isomonodromy solutions arising from irreducible SL2-local systems on the n-punctured sphere in these two cases. Concretely, each orbit type in ℒn ∪ 𝒫n determines (after choosing a compatible logarithmic connection) a finite branched covering Z → ℳ0, n such that the pullback of the universal isomonodromy connection to Z admits an algebraic horizontal section. On the pullback side this covering is essentially the relevant Hurwitz space (or a finite cover thereof) determined by the passport; on the Katz–reflection side the covering arises from the finite monodromy of the induced flat connection on the PMod0, n-orbit, equivalently from the finite image of the stabilizer in the mapping class action on the orbit set.
An important practical consequence is that our classification separates algebraic Garnier solutions into two sources with distinct geometric behaviour. Pullback-type solutions come in families parameterized by Hurwitz spaces; they are typically non-isolated in moduli and their algebraicity is visible already at the projective level through the orbifold factorization. Katz–reflection type solutions, by contrast, are rigid in the sense that they arise from isolated finite orbit points and are obtained from finite complex reflection data by an explicitly bounded Katz chain. This provides a uniform construction principle for genuinely sporadic algebraic solutions in the Garnier setting, extending the well-known role of middle convolution in the Painlev'e~VI classification.
Although our main results are formulated on the Betti side, each
certified representative (A1, …, An)
can be converted into an explicit Fuchsian system
$$
\frac{dY}{dz}=\sum_{i=1}^n \frac{R_i}{z-x_i}\,Y,\qquad
R_i\in\mathfrak{sl}_2(\overline{\mathbb{Q}}),
$$
whose monodromy representation is ρ (after a choice of framing). The
conversion is algorithmic once one fixes local exponents compatible with
the conjugacy classes of the Ai. For
Katz–reflection entries one may proceed more canonically by transporting
a rigid local system along the recorded Katz chain, using the explicit
effect of twisting and middle convolution on local exponents. For
pullback entries one may instead start from a hypergeometric (or more
generally triangle-group) system on ℙ1 \ {0, 1, ∞} and perform the
rational pullback along the stored map g, followed by the finite-order
twist needed to lift projectively.
In both cases, once one has explicit residues Ri over a number field, the Schlesinger equations give a flat algebraic connection on the parameter space of ordered poles. The finiteness of the PMod0, n-orbit implies that the corresponding horizontal section descends to an algebraic curve (or higher-dimensional algebraic subvariety in the pullback case) in the moduli of connections. Thus, beyond existence, the tables provide effective input for writing explicit algebraic Garnier solutions, at least in principle. We do not pursue this conversion systematically here, since it is primarily an exercise in symbolic computation and normalization choices, but the certification format in Appendix~A is designed so that such conversions can be carried out without ambiguity from the stored number-field data.
The mapping class action on trace-coordinate charts is given by polynomial automorphisms with integer coefficients (coming from the action of Out(π1) on words). It follows that if [ρ] has finite orbit, then all trace coordinates of [ρ] are algebraic numbers, and indeed lie in a number field stable under the induced action. This elementary observation yields useful arithmetic pre-filters for detecting candidate finite orbits before any orbit enumeration is attempted.
First, for a point given over $\overline{\mathbb{Q}}$, reduction modulo primes of good reduction produces a well-defined orbit in the corresponding character variety over finite fields, on which the mapping class generators act by the same integer polynomials. A finite orbit over ℂ necessarily reduces to a periodic orbit modulo almost all primes. Conversely, if one observes (by random reductions) that the mod-p orbit sizes grow rapidly with p, this provides strong evidence against finiteness over ℂ. While this is not a proof in either direction, it is an effective heuristic that substantially narrows the search space in computer-assisted enumeration.
Second, in the Katz–reflection regime we can predict stringent constraints on the field of definition. Since the reduced monodromy group is finite and generated by pseudoreflections with nontrivial eigenvalue λ a root of unity, the reduced traces and determinants lie in cyclotomic extensions. Tracking the Katz chain shows that the final SL2 traces lie in a controlled extension built from cyclotomic data and the character field of the relevant Shephard–Todd group. In practice, this yields an explicit upper bound on [K : ℚ] for each reflection datum, and hence bounds the complexity of exact arithmetic needed for certification. This mechanism explains why the certified enumeration remains feasible for n = 5, 6: one never encounters uncontrolled transcendental parameters on the isolated side.
Third, for points presented numerically (e.g. arising from analytic continuation of a differential equation), one can attempt an step: compute approximate trace coordinates, guess minimal polynomials by lattice reduction, and then verify the guessed algebraic point by exact substitution into the defining equations of the character variety chart and by checking the product relation. Lemma~6 then upgrades the computation to a proof of finiteness of the orbit, provided closure under the mapping class generators can be checked exactly. We emphasize that this is a verification protocol rather than a discovery method: it requires a candidate point of sufficiently low height to be recognizable, which is consistent with the bounded nature of our lists for n = 5, 6.
The dichotomy in Theorem~A is specific to n ∈ {5, 6} only in its form; conceptually it suggests a general strategy for arbitrary n in rank 2. One expects that non-isolated finite-orbit phenomena should continue to be governed by orbifold pullbacks (as in Corlette–Simpson and Loray–Pereira–Touzet), hence by Hurwitz-type data. The isolated regime should remain accessible to Katz operations: middle convolution and twisting provide a mechanism for transforming an isolated finite orbit into a local system whose local monodromies have strong eigenvalue constraints, often forcing the appearance of pseudoreflections after a suitable reduction step.
What genuinely changes for n ≥ 7 is that the reduced rank is n − 2 ≥ 5, so the reflection-group side ceases to be controlled by a finite, explicitly classified list in the same way. While irreducible finite complex reflection groups are still classified in all dimensions, the number and complexity of groups and of their pseudoreflection conjugacy classes increase quickly, and the space of admissible local data grows accordingly. Moreover, establishing an effective bounded reduction statement analogous to Proposition~D becomes subtler: the combinatorics of eigenvalue collisions under middle convolution and the geometry of fixed loci in Y(0, n, 2) both become more complicated as n increases. We therefore view the n = 5, 6 results as establishing a proof-of-concept: isolation can be exploited to force a finite, checkable reduction, but new ideas are required to keep the reduction uniformly bounded for larger n.
Finally, we indicate how the present methods fit into higher-rank questions. The basic objects (character varieties, mapping class actions, and finite orbit problems) exist verbatim for SLr with r > 2, and the relevance to algebraic isomonodromy persists. However, several features used implicitly in our rank 2 treatment fail in higher rank: trace coordinate charts are less economical, reducibility loci have more complicated stratifications, and middle convolution changes rank in a way that is harder to control. On the other hand, the philosophy behind Katz–reflection type remains meaningful: rigid or isolated finite-orbit points should be related, after explicit functorial operations, to local systems with finite monodromy generated by elements with large fixed spaces (the higher-rank analogue of pseudoreflections). One may therefore hope for a higher-rank analogue in which isolated finite orbits are reduced to finite linear groups with controlled generation, while non-isolated finite orbits are governed by orbifold or Shimura-type factorization phenomena.
From a practical perspective, the certification paradigm used here should remain applicable: any proposed classification in higher rank will inevitably be case-heavy, and isolating the computer-dependent steps as explicit, machine-checkable lemmas is essential for mathematical reliability. Appendix~A formalizes this separation in the present context; we expect the same structure to be indispensable in future extensions, even if the groups and moduli spaces involved are substantially more complicated.
This appendix records the verification protocol used in the
computer-assisted parts of Theorems~B and~C. The mathematical input is a
tuple of matrices
$$
\mathbf{A}=(A_1,\dots,A_n)\in
\mathrm{SL}_2(\overline{\mathbb{Q}})^n,\qquad \prod_{i=1}^n A_i=I,
$$
considered up to simultaneous conjugacy, together with the standard
Hurwitz (braid) action. The goal is to , in the sense of Lemma~6, that
the corresponding point in the character variety has finite PMod0, n-orbit and lies
in the irreducible locus.
We emphasize that our use of computation is not as an oracle. Every computer-dependent claim used in the main text is backed by a finite certificate consisting of explicit algebraic numbers and explicit braid words, and the verification reduces to deterministic checks in exact arithmetic over a specified number field.
A certificate begins by fixing a number field K = ℚ(α) given by a monic irreducible polynomial f ∈ ℤ[t] and a choice of embedding into ℂ only for the purpose of human-readable approximations. All algebraic data are stored and verified in K (or in a specified finite extension of K, again given by an irreducible polynomial).
Each matrix entry is represented in the power basis 1, α, …, αd − 1 with coefficients in ℚ, stored as reduced fractions. All matrix computations (multiplication, inversion, determinant) are performed exactly in this representation. When cyclotomic fields suffice, we may instead store elements in a standard ℚ(ζm) presentation; verification is nevertheless reduced to the same primitive-element model.
Given A = (A1, …, An), the first deterministic checks are:These checks ensure that A defines a point of Y(0, n, 2) on the intended local monodromy stratum.
We implement mapping class dynamics via the braid group action on
tuples with fixed product. For the Artin generator σi ∈ Bn
(1 ≤ i ≤ n − 1), the
right Hurwitz move is
σi ⋅ (A1, …, Ai, Ai + 1, …, An) = (A1, …, Ai + 1, Ai + 1−1AiAi + 1, …, An),
and σi−1
acts by the inverse transformation. This action preserves the product
constraint.
For pure mapping classes we use the standard pure braid
generators
βi, j := σj − 1⋯σi + 1 σi2 σi + 1−1⋯σj − 1−1 (1 ≤ i < j ≤ n),
which generate the pure braid group Pn. Passing from
Pn to
PMod0, n introduces
the spherical relation (the full twist becomes trivial on the sphere).
In practice, for the purpose of certifying finiteness of an orbit in the
character variety, it suffices to choose a finite generating set 𝒢 of a finite-index subgroup of PMod0, n and check
closure under 𝒢; finiteness under a
finite-index subgroup implies finiteness under the full group. Our
certificates specify such a set 𝒢
explicitly as braid words in the σi±1,
together with a short proof in the form of a reference to a standard
presentation or an explicit index computation (performed once and
recorded).
Orbit enumeration requires a robust equality test for points of Y(0, n, 2), i.e. for tuples
modulo simultaneous conjugation in SL2(K). We avoid relying
on trace coordinates alone, since for n = 5, 6 different tuples may share
many low-word traces. Instead, given two tuples A = (Ai)
and B = (Bi),
we decide whether they are simultaneously conjugate by solving for a
matrix P ∈ Mat2 × 2(K)
satisfying
PAi = BiP for
all i,
as a linear system in the four entries of P. The solution space is a K-vector space; we compute a basis
exactly. The tuples are simultaneously conjugate in GL2(K) if and only if
this space contains an invertible matrix; invertibility is tested by
checking whether det (P) ≠ 0
for some solution P, again
exactly in K. For SL2-conjugacy one may additionally
normalize by scaling P to
determinant 1, which is possible
whenever P is invertible. This
test is deterministic and does not require case distinctions on
eigenvalues.
For efficiency in large enumerations, certificates may also store a invariant (a finite list of traces of specified words) used only to pre-partition candidate equalities; the authoritative equality decision remains the linear-system test above.
Given a tuple A and a chosen generator set 𝒢 ⊂ PMod0, n (expressed as braid words), we compute its orbit by a breadth-first search in the space of tuples, with the following conventions:The stored data provide a proof certificate that the computed set S = {A(k)} is 𝒢-stable in the quotient. Verification of closure then reduces to finitely many exact matrix identities.
Formally, the certificate for finiteness consists of:Given such data, a verifier checks that every element of S satisfies the determinant and product relations, that S contains A(0), and that the closure relations hold. This implies that the ⟨𝒢⟩-orbit is contained in S, hence finite. When ⟨𝒢⟩ has finite index in PMod0, n, finiteness under PMod0, n follows.
For each representative tuple A(k) ∈ S we certify irreducibility of the associated representation ρ(k) : π1(X) → SL2(K). In rank 2 over a field of characteristic 0, reducibility is equivalent to simultaneous triangularizability, equivalently to the existence of a common invariant line in K2.
We use a purely algebraic test: compute the K-subalgebra 𝒜 ⊂ Mat2 × 2(K) generated by {Ai(k)}. Starting from the spanning set {I}, we iteratively close under multiplication by the generators and take K-linear span until stabilization. The representation is irreducible if and only if dimK(𝒜) = 4, i.e. 𝒜 = Mat2 × 2(K). Indeed, if the tuple preserves a line then all matrices are simultaneously upper triangular in a suitable basis and dimK(𝒜) ≤ 3; conversely, if dimK(𝒜) < 4 then 𝒜 is contained in a proper subalgebra, which in dimension 2 forces the existence of an invariant line. This test is stable under field extensions and avoids special handling of repeated eigenvalues.
In certificates we typically record, for each orbit S, a single irreducibility proof for one representative together with the observation that irreducibility is preserved by Hurwitz moves and conjugation; nevertheless, the verifier may (and in our reference scripts does) re-check irreducibility for every element of S to guard against implementation errors.
On the Katz–reflection side, certificates include in addition a chain of operationsIn particular, when we claim that the reduced generators are pseudoreflections, we check directly that each has eigenvalue 1 of multiplicity n − 3 and a single nontrivial eigenvalue λ ≠ 1, equivalently that rank(Bi − I) = 1.
For each orbit type listed in the appendices, we provide: the defining polynomial of K, the explicit matrices Ai, the generating words for 𝒢, the orbit set S (or a compressed form sufficient to reconstruct it), and the conjugators Pk, g. A verification script, written to avoid heuristic steps (no floating point, no randomized choices), checks the identities described in A.1–A.6. Since all checks reduce to finitely many equalities in K, the verification can be run in any system supporting exact number field arithmetic.
We regard this separation between (i) discovery/enumeration and (ii) certificate verification as essential. Enumeration may use optimizations, memoization, and preliminary filters, but the mathematical claims used in the main text depend only on the existence of a valid certificate, which can be checked independently and mechanically.
We record the conventions for Katz’s middle convolution in the
multiplicative (Betti) setting, and the concrete linear-algebraic model
used to produce exact matrix representatives in the certificates of
Appendix~. Throughout, X = ℙ1 \ {x1, …, xn}
with n ∈ {5, 6}, and a rank
r local system on X is represented by a tuple
A = (A1, …, An) ∈ GLr(K)n, A1⋯An = I,
for some number field K, where
Ai is the
monodromy of γi in the fixed
presentation of π1(X).
Fix λ ∈ ℂ× \ {1}. The
multiplicative middle convolution MCλ is defined as
convolution with a rank-one Kummer local system of parameter λ, followed by the middle extension
and restriction back to X. For
computations on tuples we encode the Kummer system by a rank-one
character κλ : π1(X) → ℂ×
with prescribed local monodromies. Concretely, we choose the standard
normalization
so that ∏iκλ(γi) = 1
and twisting preserves the relation A1⋯An = I.
Thus, if A represents
ρ, then ρ ⊗ κλ
is represented by
A(λ) := (λA1, …, λAn − 1, λ−(n − 1)An).
More generally, for any finite-order rank-one twist χ (as in the main text), represented
by scalars χi ∈ K×
with ∏iχi = 1,
we use the notation
A ⊗ χ := (χ1A1, …, χnAn).
All twists are implemented entrywise in exact arithmetic and verified by
determinant and product checks.
We use the explicit description of MCλ on monodromy representations via parabolic cohomology, as developed by Katz and made algorithmic by Dettweiler–Reiter. We briefly fix a concrete model that is sufficient for deterministic computation.
Let V = Kr
be the representation space of ρ ⊗ κλ,
so the local monodromies are Ai(λ).
A (normalized) 1-cocycle for π1(X) with
values in V is determined by
its values
vi := b(γi) ∈ V, i = 1, …, n,
subject to the cocycle relation coming from γ1⋯γn = 1:
The condition is
and the coboundaries are those tuples of the form
The (multiplicative) middle convolution space is then realized as the
quotient
This construction is functorial in ρ, hence carries a natural action of
π1(X)
induced from the action on cocycles; by definition MCλ(ρ) is the
resulting representation on Wλ(ρ),
after discarding the maximal trivial subrepresentation (equivalently,
passing to the middle extension, which in this model amounts to the
parabolic condition ).
The resulting dimension dimK(W) is the computed rank of MCλ(ρ). In the cases relevant to Proposition~D, this rank is n − 2, yielding dimension 3 for n = 5 and 4 for n = 6.
To obtain explicit monodromy matrices on W, we use the standard action of
π1(X) on
1-cocycles:
(g ⋅ b)(h) = ρλ(g) b(g−1hg), ρλ := ρ ⊗ κλ,
which preserves the parabolic subspace and coboundaries, hence descends
to W. Since π1(X) is
generated by the γi, it suffices
to compute the induced linear operators of each γi on a chosen
basis of W. Algorithmically,
for each basis class [v] ∈ W represented by
v = (v1, …, vn) ∈ Z,
we compute a representative of γi ⋅ [v]
by applying the above formula to cocycles and then reducing modulo B. This yields matrices
B = (B1, …, Bn) ∈ GL(W)n, B1⋯Bn = I,
representing MCλ(ρ) with
respect to the chosen basis of W. (The product relation holds
formally because it holds in π1(X); we
nevertheless verify it as an exact identity in K for each certificate.)
This approach avoids any ambiguity from choosing generators of intermediate free groups: everything is computed directly from the fixed presentation ⟨γ1, …, γn ∣ ∏γi = 1⟩ used throughout the manuscript.
Middle convolution is naturally GL-valued. When we need SL2-valued data (before applying MC) or need to compare outputs up to scalar twists, we normalize using rank-one twists as follows.
On the input side, our tuples in the main classification live in SL2. When applying MCλ we first allow twisting by a finite-order χ to arrange the eigenvalue pattern required by Lemma~4; this does not leave the SL2-world because χiI2 has determinant χi2, and we compensate by choosing χ of order dividing a suitable even integer so that χi2 = 1 when required. On the output side, when comparing with a Shephard–Todd reflection group in GLn − 2, we keep the natural GL-representation: reflection groups are not constrained to determinant 1, and in fact the pseudoreflection eigenvalue λ is tracked via determinants.
For bookkeeping, we record for each output generator Bi the
characteristic polynomial
pi(t) = det (tI − Bi) ∈ K[t]
and the determinant det (Bi) = pi(0).
This is the data used in Appendix~C to match the reduced tuple with a
specified pseudoreflection class.
In dimensions 3 and 4, a matrix R ∈ GLm(K) is a pseudoreflection if and only if rank(R − I) = 1, equivalently dim ker (R − I) = m − 1. In certificates arising from Proposition~D we verify pseudoreflection-ness using the rank test over K, which is deterministic.
When we additionally want to read off the nontrivial eigenvalue, we
compute it as the unique scalar η ≠ 1 such that
det (R) = η,
since the eigenvalues are 1, …, 1, η. For the Katz–reflection
reductions in the main text, after the appropriate choice of λ and twist χ, this eigenvalue equals the
convolution parameter (up to the fixed convention , i.e. up to replacing
λ by λ−1 depending on whether
we apply MCλ or
MCλ−1).
On the strata we consider, MCλ is an equivalence up to the expected rank-one twists, and we use it as an invertible move in the equivalence relation on local systems. Concretely, whenever MCλ(ρ) has the expected rank and no trivial subquotients, we verify that applying the same construction with parameter λ−1 returns a tuple equivalent to the original one, after an explicit twist that can be read off from determinants. In computations this is checked by constructing both tuples and testing simultaneous conjugacy in the relevant GL-group (as in Appendix~, A.3).
Because W = Z/B is constructed by linear algebra, its basis is not canonical. Our certificates therefore store:A verifier reconstructs Z, B, and W from A, λ, and χ, checks that the stored basis indeed maps to a basis of W, and then recomputes the induced γi-actions to confirm the stored Bi. This keeps the middle convolution step within the same exact arithmetic framework as the braid-orbit checks of Appendix~.
Finally, we note that in all instances used in Theorems~B and~C the parameter λ is a root of unity and the input tuples are defined over number fields stable under adjoining λ. Accordingly, we choose K to contain the necessary cyclotomic subfield from the outset, so that no field extension is introduced mid-certificate. This simplifies independent verification and ensures that all identities reduce to exact equalities in a single specified number field.
In this appendix we describe the finite-group enumeration that
underlies the Katz–reflection side of Theorems~B and~C. Concretely, for
n ∈ {5, 6} we set
m := n − 2 ∈ {3, 4}.
By Proposition~D and Lemma~4, an isolated finite orbit point [ρ] ∈ Y(0, n, 2)irr
admits a Katz reduction to a GLm-local system whose
local monodromies at n − 1
punctures are pseudoreflections. The purpose of the present appendix is
to explain how we filter the Shephard–Todd list in dimensions m = 3, 4 to those finite complex
reflection groups that can arise, and how we classify the relevant
pseudoreflection tuples up to the standard equivalences.
Condition is a convenient algebraic proxy for the absence of a trivial subrepresentation in the corresponding local system; it is automatically satisfied in the instances arising from our middle convolution certificates, and it provides an inexpensive early rejection test during enumeration.
Two admissible tuples are considered equivalent if they differ by simultaneous conjugacy in G and by Hurwitz moves (i.e. the standard braid group Bn action on tuples with product 1). In the main text we package this as a (G, λ, D), where D is taken modulo this equivalence.
The computational task addressed here is therefore:
If R ∈ GLm(ℂ) is a pseudoreflection, then its eigenvalues are (1, …, 1, η) for a unique η ≠ 1, and det (R) = η. Thus in the admissibility definition above, prescribing det (Di) = λ is equivalent to prescribing the nontrivial eigenvalue of each pseudoreflection to be λ. This matches the output-side convention of Appendix~, B.5: in our certificates, after the chosen twist and convolution parameter, the reduced local monodromies at n − 1 punctures have a uniform nontrivial eigenvalue which is recorded as λ.
For a fixed G, the possible determinants of pseudoreflections form a finite subset of roots of unity and can be read off from the list of reflection conjugacy classes. In practice we proceed as follows. For each Shephard–Todd group G in dimension m, we enumerate the conjugacy classes of reflections C ⊂ G, compute det (C), and retain only those classes with det (C) ≠ 1. Each retained class contributes a candidate parameter λ := det (C) together with the constraint that Di ∈ C for i ≤ n − 1. This class-level reduction drastically limits the ensuing tuple search.
We emphasize that for m = 3, 4 there are only finitely many irreducible G and finitely many reflection classes in each; this is the content used abstractly in Lemma~5.
Fix a group G and a reflection class C ⊂ G with det (C) = λ ≠ 1. We search for (n − 1)-tuples (D1, …, Dn − 1) ∈ Cn − 1 satisfying the irreducibility and nondegeneracy conditions, and we set Dn = (D1⋯Dn − 1)−1 automatically.
A direct brute-force scan of Cn − 1 is usually unnecessary. Instead we build tuples incrementally and apply fast rejection tests:All three tests are exact linear algebra and group computations and are readily recorded in a certificate as explicit matrices and subgroup orders.
The local system associated with a reduced tuple (D1, …, Dn) is defined only up to Hurwitz equivalence (braid action) and simultaneous conjugation. Accordingly, once candidate tuples have been produced we classify them by their braid orbits.
We use the standard Hurwitz generators σi ∈ Bn
acting by
σi: (D1, …, Di, Di + 1, …, Dn) ↦ (D1, …, Di + 1, Di + 1−1DiDi + 1, …, Dn),
which preserve the product D1⋯Dn.
Since Dn
is determined by the first n − 1 factors, it is convenient to
implement the braid action on the full n-tuple but store representatives by
their first n − 1 entries.
The tuple classification above is purely group-theoretic. In the main argument we require that it be compatible with the Katz-reduced tuples arising from MCλ−1(ρ ⊗ χ−1). The compatibility checks we impose at the reflection-group stage are those invariant under the invertible Katz operations and that can be verified without reference to an ambient SL2-tuple.
The principal invariants are:In particular, we do not attempt to reconstruct the original SL2-data at this stage. That reconstruction is performed later (Appendix~) by applying inverse Katz operations to each accepted reflection datum, producing explicit SL2-tuples A whose braid orbits are then certified directly.
The end product of Appendix~ is a finite list ℒn of Katz–reflection data for n = 5, 6. Each element of ℒn is stored with enough information to permit independent verification:These objects are precisely the inputs used in the inverse Katz-operation chains that generate the SL2-tuples classified in Theorems~B and~C. The separation between the present appendix (reflection-group filtering) and Appendix~ (full braid-orbit certification in SL2) isolates the finite-group dependence: any discrepancy at the reflection level manifests as a failure of the later SL2-verification steps, and conversely each accepted datum in ℒn produces at least one certified finite orbit after applying the stored Katz chains.
Finally, we note that nothing in this appendix uses the hypothesis that the original SL2-local system has finite local monodromies; only the finiteness and isolation of the PMod0, n-orbit enters (via Proposition~D and Lemma~4), which is precisely why the reflection-group filtering is the correct replacement for the older ``finite local monodromy’’ reductions in the n = 5, 6 setting.