A curiously cubulated group

TL;DR

This paper constructs a specific 2-dimensional group that acts properly on a locally finite CAT(0) cube complex but cannot do so on any finite-dimensional CAT(0) cube complex, revealing new insights into group actions on cube complexes.

Contribution

It introduces a finitely generated 2-dimensional group with unique action properties on CAT(0) cube complexes, highlighting differences between local finiteness and finite dimensionality.

Findings

Group acts properly on locally finite CAT(0) cube complex

Group does not act properly on any finite-dimensional CAT(0) cube complex

Reveals distinctions in group actions related to cube complex dimensions

Abstract

We construct a finitely generated 2-dimensional group that acts properly on a locally finite CAT(0) cube complex but does not act properly on a finite dimensional CAT(0) cube complex.