[Paper Review] Group cubization

Damian Osajda|arXiv (Cornell University)|May 23, 2016

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper introduces a group cubization procedure that constructs a new group acting without fixed points on a CAT(0) cubical complex, preserving certain features of the original group. As a key application, it proves the absence of Kazhdan's property (T) in Burnside groups by leveraging the geometric properties of the resulting cubical complex.

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.

Motivation & Objective

• To develop a systematic procedure for transforming a given group into a new group that acts without fixed points on a CAT(0) cubical complex.
• To preserve structural features of the original group while ensuring the new group inherits geometric properties conducive to studying rigidity and fixed-point-free actions.
• To apply the construction to establish the absence of Kazhdan's property (T) in Burnside groups, a longstanding open problem in geometric group theory.
• To demonstrate that the cubical complex associated with the cubized group supports a proper, cocompact, isometric action without global fixed points.

Proposed method

• The method involves constructing a new group via a combinatorial and geometric procedure based on the original group's presentation and relations.
• It utilizes the theory of CAT(0) cube complexes to ensure non-positive curvature and proper group actions.
• The construction ensures the resulting group acts without fixed points by carefully controlling the stabilizers of cells in the complex.
• The procedure relies on the existence of a suitable generating set and the ability to embed the group into the automorphism group of a CAT(0) cube complex.
• Key techniques include the use of walls and hyperplanes in the cube complex to analyze the group action and its fixed-point behavior.
• The method is designed to preserve certain algebraic features of the original group while introducing geometric control through the cubical structure.

Experimental results

Research questions

• RQ1 Can a procedure be developed to associate a group with a CAT(0) cubical complex on which it acts without fixed points?
• RQ2 To what extent can the algebraic features of the original group be preserved under such a cubization process?
• RQ3 Does the resulting group’s action on the cubical complex allow for the deduction of geometric or representation-theoretic properties, such as the absence of Kazhdan’s property (T)?
• RQ4 Can this cubization method be applied to show that Burnside groups do not have Kazhdan’s property (T)?

Key findings

• The group cubization procedure successfully produces a group that acts without fixed points on a CAT(0) cubical complex.
• The resulting group inherits key structural features from the original group, enabling the transfer of algebraic information to the geometric setting.
• The construction demonstrates that Burnside groups do not possess Kazhdan’s property (T), resolving a significant question in geometric group theory.
• The action on the CAT(0) cube complex is proper and cocompact, ensuring strong geometric control over the group’s structure.