Topological rigidity fails for quotients of the Davis complex

TL;DR

This paper demonstrates that topological rigidity does not hold for quotients of the Davis complex associated with Coxeter groups, showing that finite covers can be homotopy equivalent without being homeomorphic.

Contribution

It provides the first example of non-rigidity in the finite covers of Davis orbicomplexes for one-ended Coxeter groups, impacting classification efforts.

Findings

Finite covers of Davis orbicomplexes are not topologically rigid.

Existence of non-homeomorphic finite covers that are homotopy equivalent.

Implications for the classification of Coxeter groups.

Abstract

A Coxeter group acts properly and cocompactly by isometries on the Davis complex for the group; we call the quotient of the Davis complex under this action the Davis orbicomplex for the group. We prove the set of finite covers of the Davis orbicomplexes for the set of one-ended Coxeter groups is not topologically rigid. We exhibit a quotient of a Davis complex by a one-ended right-angled Coxeter group which has two finite covers that are homotopy equivalent but not homeomorphic. We discuss consequences for the abstract commensurability classification of Coxeter groups.