Vietoris-Rips complexes of affine Coxeter groups

Abstract

We prove that all affine Coxeter groups admit contractible Vietoris-Rips complexes. More precisely, for any affine Coxeter group $W_a$ with standard generating set $S$, we prove that the Vietoris-Rips complex ${\mathcal{V}\mathcal{R}}_t(W_a;S)$ is contractible for all sufficiently large $t$. We also investigate other finitely generated groups of the form $G=H⋊F$ for $F$ finite, and prove that for any finite symmetric generating set $A$ for $H$ such that $A^F=A$, the complexes ${\mathcal{V}\mathcal{R}}_t(G;AF)$ and ${\mathcal{V}\mathcal{R}}_t(H;A)$ are homotopy equivalent for all $t\ge 1$.

arXiv comment19 pages, 3 figures. V2: Submitted version. The results about Coxeter groups are unchanged. The more general results are slightly weaker, following our fixing an error (a certain generating set needed to be invariant under conjugation by a certain finite subgroup)