Group cubization (with an appendix by Mikael Pichot)

TL;DR

The paper introduces a group cubization procedure that produces groups acting without fixed points on CAT(0) cubical complexes, and uses this to show Burnside groups lack Kazhdan's property (T).

Contribution

It presents a novel method to construct groups with specific geometric actions and applies it to prove Burnside groups do not have Kazhdan's property (T).

Findings

Burnside groups lack Kazhdan's property (T).

Group cubization creates groups acting on CAT(0) cubical complexes.

New technique links group actions with algebraic properties.

Abstract

We present a procedure of group cubization: It results in a group whose some features resemble the ones of a given group, and which acts without fixed points on a CAT(0) cubical complex. As a main application we establish lack of Kazhdan's property (T) for Burnside groups.