Coxeter groups, quiver mutations and geometric manifolds

TL;DR

This paper constructs hyperbolic manifolds with large symmetry groups using Coxeter groups, quiver mutations, and cluster algebra techniques, generalizing to various types of quivers and diagrams.

Contribution

It introduces a novel method to build hyperbolic manifolds via quiver mutations and extends the construction to diverse quiver types and geometric structures.

Findings

Constructed finite volume hyperbolic manifolds with large symmetry groups.

Developed a CW-complex framework compatible with quiver mutations.

Extended the construction to quivers from surfaces and orbifolds.

Abstract

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action of a finite (or affine) Coxeter group. These CW-complexes undergo mutations agreeing with mutations of quivers and diagrams. We also generalize the construction to quivers and diagrams originating from unpunctured surfaces and orbifolds.