A topologically rigid set of quotients of the Davis complex

TL;DR

This paper investigates the topological rigidity of quotients of the Davis complex associated with right-angled Coxeter groups, identifying conditions that prevent rigidity and constructing infinitely many rigid subclasses.

Contribution

It provides new conditions on defining graphs that obstruct topological rigidity and introduces infinitely many topologically rigid subclasses of Davis complex quotients.

Findings

Certain graph conditions prevent topological rigidity

Many subclasses of Davis complex quotients are topologically rigid

Topological rigidity is rare but achievable in specific cases

Abstract

A class of topological spaces is topologically rigid if any two spaces with the same fundamental group are also homeomorphic. Topological rigidity, in addition to its intrinsic interest, has been useful for solving abstract commensurability questions. In this paper, we explore the topological rigidity of quotients of the Davis complex of certain right angled Coxeter groups by providing conditions on the defining graphs that obstruct topological rigidity. Furthermore, we explore why topological rigidity is hard to achieve for quotients of the Davis complex. Nonetheless, we conclude by introducing infinitely many infinite topologically rigid subclasses.