CAT(0) cube complexes with flat hyperplanes

TL;DR

This paper proves that groups acting on CAT(0) cube complexes with virtually abelian hyperplane-stabilisers have a specific algebraic decomposition into free products of free abelian and surface groups.

Contribution

It establishes a new structural decomposition result for groups acting on CAT(0) cube complexes under certain stabilizer conditions.

Findings

Groups decompose as free products of free abelian and surface groups.

Hyperplane-stabilizers being virtually abelian implies a specific group structure.

Provides insight into the algebraic structure of groups acting on CAT(0) cube complexes.

Abstract

In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.