Cocompactly cubulated crystallographic groups

TL;DR

This paper characterizes the simplicial boundary of cocompactly cubulated crystallographic groups as hyperoctahedral triangulations of spheres, linking group properties with geometric boundary structures.

Contribution

It establishes that cocompactly cubulated crystallographic groups are exactly the hyperoctahedral groups, connecting group theory with geometric and topological boundary analysis.

Findings

The simplicial boundary is isomorphic to the hyperoctahedral triangulation of $S^{n-1}$.

Cocompactly cubulated crystallographic groups are precisely hyperoctahedral groups.

Provides an answer to Wise's question on cubulating virtually free abelian groups.

Abstract

We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually $\integers^n$ group is isomorphic to the hyperoctahedral triangulation of $S^{n-1}$, providing a class of groups $G$ for which the simplicial boundary of a $G$-cocompact cube complex depends only on $G$. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension $n$ are precisely those that are \emph{hyperoctahedral}. We apply this result to answer a question of Wise on cocompactly cubulating virtually free abelian groups.