Punctured Weyl Groups in Marked Surfaces

• Punctured Weyl groups are algebraic structures derived from triangulated punctured surfaces that yield mutation-invariant quotients of Coxeter groups.
• They arise via quiver mutations on marked surfaces where additional cycle relations extend classical Weyl presentations to include puncture and orbifold phenomena.
• In higher Teichmüller theory, local Weyl group actions at punctures structure cluster varieties through tensor-diagram invariants and marked-surface moduli.

Searching arXiv for recent and foundational papers on punctured surfaces, Weyl groups, and related cluster/Coxeter constructions. “Punctured Weyl groups” is not a standard single term in the literature. In current research, it most naturally denotes two distinct but closely related families of constructions. One family assigns mutation-invariant groups to triangulations of punctured marked surfaces and orbifolds; these groups are presented as quotients of Coxeter groups and extend the Barot–Marsh program from Dynkin and affine mutation classes to most punctured surfaces (Felikson et al., 2024). The other family concerns puncture-local Weyl group actions on cluster varieties and higher Teichmüller spaces, where each puncture contributes a copy of the Weyl group acting on the local decoration data (Inoue et al., 2019, Fraser et al., 2021). A narrower, more classical interpretation views “puncturing” as deleting a Dynkin node and studying how a rank $n-1$ parabolic subgroup is completed back to the full Weyl group (Baumeister et al., 2017). By contrast, the theory of punctured spheres via welded graphs and Wirtinger presentations concerns complement groups and peripheral data rather than Weyl groups (Audoux et al., 2023).

1. From Dynkin mutation classes to punctured surfaces

The punctured-surface theory grows out of the Barot–Marsh construction of a finite Weyl group from any quiver mutation-equivalent to an orientation of a Dynkin diagram, and its later extension to affine Coxeter groups and to groups arising from triangulations of unpunctured surfaces and orbifolds (Felikson et al., 2013). In that earlier framework, one starts from a Coxeter presentation attached to the underlying quiver or diagram and imposes additional relations coming from oriented cycles and certain distinguished local configurations. The resulting group is invariant under mutation, even though the ambient Coxeter presentation changes with the quiver.

The punctured extension retains that philosophy but introduces new phenomena caused by punctures. In the punctured case, loop-free triangulations can produce long oriented cycles coming from once-punctured polygons or discs, and more complicated local configurations appear when self-folded triangles or orbifold points are allowed. The main theorem applies to marked surfaces with at least $4$ features, where a feature means either a puncture or a boundary component, and states that the resulting group $G(Q)$ is an invariant of the marked surface $S$, independent of the loop-free triangulation used to define the quiver $Q$ (Felikson et al., 2024).

This punctured theory is therefore best understood as a generalization of finite and affine Weyl-group presentation theory from mutation classes of quivers to marked surfaces. The surface no longer yields a Weyl group in general. Instead, it carries a canonical quotient of many different Coxeter groups, one for each loop-free triangulation in the mutation class.

2. Punctured-surface groups as quotients of Coxeter groups

Let $Q$ be the quiver of a loop-free triangulation of a punctured marked surface. The associated group $G(Q)$ is generated by $s_1,\dots,s_n$, one generator per vertex of $Q$. In the basic loop-free punctured-surface case, the defining relations are:

$s_i^2=e \qquad \text{for all } i,$

together with Coxeter-type relations

$4$0

where $4$1 if $4$2 and $4$3 are not joined and $4$4 if they are joined by an arrow, and cycle relations attached to every chordless oriented cycle in $4$5 (Felikson et al., 2024). In this basic setting there are no separately named “puncture relations” or “handle relations”; the only defining relations are the involution relations, the Coxeter exponents $4$6, and the cycle relations.

The group is explicitly a quotient of an ambient Coxeter group. If $4$7 denotes the Coxeter group generated by the same $4$8 with only the involution and Coxeter-type relations, then

$4$9

where $G(Q)$0 is the normal closure of the additional cycle relations, and in more general settings also of the relations required for self-folded triangles or orbifold blocks (Felikson et al., 2024). For orbifolds with order-$G(Q)$1 orbifold points, arrows of weight $G(Q)$2 appear and the Coxeter exponent $G(Q)$3 is added. If triangulations with self-folded triangles are allowed, one must further impose the relations of type $G(Q)$4; in the orbifold setting these become $G(Q)$5.

This formulation explains the phrase “punctured Weyl groups” in a precise algebraic sense. The punctured surface does not canonically produce a Weyl group, but it does canonically produce a mutation-invariant quotient of Coxeter groups associated with triangulation quivers. In finite Dynkin mutation classes these quotients recover ordinary Weyl groups; in the punctured-surface case they need not.

3. Mutation invariance, triangulations, and representative examples

The invariance mechanism has two levels. Locally, a loop-free flip $G(Q)$6 corresponding to mutation at a vertex $G(Q)$7 induces an isomorphism

$G(Q)$8

where $G(Q)$9. The explicit transformed generators are

$S$0

Globally, one proves that when the marked surface has at least $S$1 features, any two loop-free triangulations are connected by a sequence of loop-free flips up to combinatorial equivalence, so the isomorphism type of $S$2 depends only on the surface (Felikson et al., 2024).

The punctured theory is broad but not universal. The paper excludes general surfaces and orbifolds with fewer than $S$3 features. If the only features are punctures and there are too few of them, there may be no loop-free triangulation. In the $S$4-feature case, loop-free flip connectivity can fail; a counterexample is given for a thrice punctured genus $S$5 surface. The once punctured surface or orbifold with one boundary component remains unresolved (Felikson et al., 2024).

Two examples are structurally central. The once-punctured annulus with one marked point on each boundary component is described as “the minimal example of a punctured surface with the corresponding quiver of non-finite and non-affine type.” Its group $S$6 is a quotient of a $S$7-generator Coxeter group and admits a surjective homomorphism onto the affine Weyl group of type $S$8, but the homomorphism is not an isomorphism; the kernel is nontrivial. This shows that punctured-surface groups can surject onto affine Weyl groups while being genuinely new objects (Felikson et al., 2024).

A second family comes from a closed surface of genus $S$9 with three punctures. A loop-free triangulation yields an oriented cycle of length $Q$0, so the group presentation has generators $Q$1, Coxeter relations $Q$2 cyclically and $Q$3 for nonadjacent vertices, together with a single cycle relation

$Q$4

This is a direct punctured-surface analogue of the Barot–Marsh construction (Felikson et al., 2024).

4. Weyl groups acting at punctures in higher Teichmüller theory

A different meaning of “punctured Weyl groups” arises in cluster and higher Teichmüller theory. For a symmetrizable Kac–Moody Lie algebra $Q$5, one constructs weighted quivers $Q$6 whose cluster modular group contains the Weyl group $Q$7 as a subgroup. The corresponding simple reflections are realized by explicit mutation sequences $Q$8, and the induced $Q$9- and $Q$0-transformations are given by closed formulas (Inoue et al., 2019).

For classical finite type with Coxeter number $Q$1, the quiver $Q$2 is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with $Q$3 marked points on the boundary, up to frozen vertices. This identifies the abstract cluster realization of $Q$4 with a genuinely punctured-surface construction. More generally, for a marked surface $Q$5 with $Q$6 punctures, the paper proves that there is a canonical embedding

$Q$7

and that the resulting cluster action on $Q$8 coincides with the geometric action of Goncharov–Shen (Inoue et al., 2019).

For $Q$9, the puncture-local Weyl group is $G(Q)$0. At each puncture $G(Q)$1, the monodromy $G(Q)$2 stabilizes an affine flag $G(Q)$3, and the simple reflection $G(Q)$4 acts by replacing only the $G(Q)$5-th step of that flag. This yields birational automorphisms $G(Q)$6 of the moduli space, and the actions at different punctures commute, giving

$G(Q)$7

On initial cluster variables $G(Q)$8 adjacent to a puncture $G(Q)$9, the action is row-local:

$s_1,\dots,s_n$0

and

$s_1,\dots,s_n$1

where $s_1,\dots,s_n$2 is the puncture potential expressed as a sum of rhombus monomials. Goncharov–Shen’s theorem shows that, except in certain exceptional $s_1,\dots,s_n$3 cases, this puncture Weyl action is by cluster automorphisms; the same holds for the Grassmannian version $s_1,\dots,s_n$4, and for $s_1,\dots,s_n$5 also on $s_1,\dots,s_n$6 (Fraser et al., 2021).

5. Tagging, tensor diagrams, and higher-rank puncture combinatorics

In rank $s_1,\dots,s_n$7, puncture combinatorics is governed by tagged arcs and tagged triangulations: each arc end at a puncture is plain or notched. The higher-rank replacement is not a binary tag but a Weyl-group orbit of puncture labels. For $s_1,\dots,s_n$8, a leg of weight $s_1,\dots,s_n$9 at a puncture can be tagged by any subset

$Q$0

which lies in the $Q$1-orbit of the fundamental weight $Q$2. Thus the puncture-local combinatorics is controlled by $Q$3, not by a single involution (Fraser et al., 2021).

This idea is formalized through tensor diagrams. A pseudotagging is a function

$Q$4

such that $Q$5. A tagged tensor diagram imposes the puncturewise nesting condition: for any two legs $Q$6 at the same puncture,

$Q$7

Tagged tensor diagram invariants are exactly the pullbacks of ordinary diagram invariants along the puncture Weyl action. In particular, if a tagged diagram is obtained by applying one Weyl element $Q$8 at each puncture $Q$9, then its invariant is the pullback by $s_i^2=e \qquad \text{for all } i,$0 of the untwisted invariant (Fraser et al., 2021).

For $s_i^2=e \qquad \text{for all } i,$1, this reproduces the classical tagged-arc picture exactly. The two labels $s_i^2=e \qquad \text{for all } i,$2 and $s_i^2=e \qquad \text{for all } i,$3 correspond to plain and notched, and the nontrivial simple reflection $s_i^2=e \qquad \text{for all } i,$4 interchanges them. The paper gives an explicit once-punctured digon computation in which the notched variable $s_i^2=e \qquad \text{for all } i,$5 is obtained from the plain diagram invariant by the puncture Weyl action.

A central algebraic result is the flattening theorem: every pseudotagged diagram invariant is a linear combination of tagged diagram invariants. This is the higher-rank replacement for resolving incompatible taggings. It shows that products of puncture-Weyl-tagged cluster data, although not themselves tagged in the strict sense, can still be rewritten in the tagged basis. This suggests that the correct higher analogue of tagged triangulations is a Weyl-tagged tensor-diagram calculus rather than a direct combinatorics of arcs alone (Fraser et al., 2021).

6. Adjacent interpretations and terminological limits

The phrase “punctured Weyl groups” also admits a narrower finite-type interpretation based on deleting a simple generator. For a finite crystallographic Weyl group $s_i^2=e \qquad \text{for all } i,$6, if $s_i^2=e \qquad \text{for all } i,$7 is a rank $s_i^2=e \qquad \text{for all } i,$8 parabolic subgroup, then every reflection $s_i^2=e \qquad \text{for all } i,$9 with

$4$00

lies in a single $4$01-conjugacy orbit. Equivalently, after deleting one simple root, any reflection restoring the full group is $4$02-conjugate to the omitted simple reflection. The same paper proves a lattice criterion: a reflection set $4$03 generates $4$04 iff its roots and coroots generate the full root and coroot lattices. This is a deleted-node or near-parabolic notion of puncturing, rather than a punctured-surface one (Baumeister et al., 2017).

There are also broader chamber-restriction constructions that are Weyl-theoretic but not punctured in the surface sense. For a strongly dominant weight $4$05, the dominant weight polytope

$4$06

is a fundamental region for the $4$07-action on the $4$08-permutohedron $4$09, and the fixed-subring theorem

$4$10

identifies the cohomology of the chamber-restricted toric orbifold with the invariant cohomology of the full Weyl-chamber toric variety (Gui et al., 2024). This is a quotient or folding phenomenon, not a puncture construction.

Finally, the term should not be extended to punctured-sphere topology. The paper on welded graphs, Wirtinger groups, and knotted punctured spheres explicitly states that the relevant algebraic structure is not a Coxeter or Weyl group. Its groups are Wirtinger presentations and fundamental groups of exteriors of ribbon surface-links, with punctured spheres encoded by welded forests and classified up to link-homotopy by non-repeated Milnor invariants (Audoux et al., 2023).

Taken together, these developments suggest that “punctured Weyl groups” is best used as an umbrella label for Weyl- and Coxeter-theoretic structures localized at punctures or produced from punctured surfaces, rather than as the name of a single universal object. In the strongest current sense, the phrase refers either to mutation-invariant Coxeter quotients attached to punctured marked surfaces or to puncture-local Weyl group actions on cluster and higher Teichmüller moduli spaces.