Punctured surfaces, quiver mutations, and quotients of Coxeter groups

TL;DR

This paper extends the construction of Coxeter group presentations from quiver mutations to include punctured surfaces and orbifolds, offering new invariants for these topological objects.

Contribution

It generalizes previous work to cover punctured surfaces and orbifolds, creating new invariants and broadening the applicability of Coxeter group presentations.

Findings

Constructed Coxeter group presentations for punctured surfaces.

Provided invariants for marked surfaces and orbifolds.

Extended quiver mutation methods to new topological cases.

Abstract

In 2011, Barot and Marsh provided an explicit construction of presentation of a finite Weyl group $W$ by any quiver mutation-equivalent to an orientation of a Dynkin diagram with Weyl group $W$. The construction was extended by the authors of the present paper to obtain presentations for all affine Coxeter groups, as well as to construct groups from triangulations of unpunctured surfaces and orbifolds, where the groups are invariant under change of triangulation and thus are presented as quotients of numerous distinct Coxeter groups. We extend the construction to include most punctured surfaces and orbifolds, providing a new invariant for almost all marked surfaces.