Topological rigidity fails for quotients of the Davis complex

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 oneended 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.

A topologically rigid set of quotients of the Davis complex

“…For further background on the Davis complex and Coxeter groups, refer to [3]. The Davis orbicomplex has been studied by Stark, who poses the following question in [7], which we will give a partial answer to in this paper: Question 1.2. For which set W of Coxeter groups is the set of Davis orbicomplexes D Γ for groups in W together with their finite-sheeted covers topologically rigid?…”

“…Let Γ be a cycle of generalized Θ graphs with Davis complex Σ Γ , and G a finite index, torsionfree subgroup of W Γ . Stark proves in [7] that the set of quotients Σ Γ /G, which correspond to finite-sheeted covers of the Davis orbicomplexes D Γ , is not topologically rigid by constructing X 1 = Σ Γ /G 1 and X 2 = Σ Γ /G 2 that are homotopic but not homeomorphic. Theorem 1.7 in Section 2 generalizes the construction from [7] to create a class of orbicomplexes where topological rigidity fails.…”

“…Stark proves in [7] that the set of quotients Σ Γ /G, which correspond to finite-sheeted covers of the Davis orbicomplexes D Γ , is not topologically rigid by constructing X 1 = Σ Γ /G 1 and X 2 = Σ Γ /G 2 that are homotopic but not homeomorphic. Theorem 1.7 in Section 2 generalizes the construction from [7] to create a class of orbicomplexes where topological rigidity fails. Our construction of homotopic but not homeomorphic covers relies on the fact that one set of orbifolds in the orbicomplex is a finite cover of another set of orbifolds.…”

Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

“…The natural spaces to apply this strategy to in our setting are quotients Σ Γ /G where Σ Γ is the Davis complex for W Γ , and G is a torsion-free, finite-index subgroup of W Γ . However, Stark proves in [22] that topological rigidity fails for such quotients, by constructing an example where G and G are isomorphic torsion-free, finite-index subgroups of W Γ , but Σ Γ /G and Σ Γ /G are not homeomorphic. The graph Γ in this example is a 3-convex cycle of generalized Θ-graphs.…”

“…In light of the result of [22], in Section 3 we introduce a new geometric realization for right-angled Coxeter groups W Γ with 3-convex Γ ∈ G, by constructing a piecewise hyperbolic orbicomplex O Γ with fundamental group W Γ . The orbicomplex O Γ has underlying space obtained by gluing together right-angled hyperbolic polygons, and each edge of O Γ which is contained in only one such polygon is a reflection edge.…”