Geometric dimension of groups for the family of virtually cyclic subgroups

TL;DR

This paper investigates the geometric dimension of groups with respect to virtually cyclic subgroups, providing bounds, criteria, and examples that deepen understanding of classifying spaces in geometric group theory.

Contribution

It establishes bounds on the geometric dimension for elementary amenable groups, offers criteria for extensions with torsion-free quotients, and presents examples answering a question by W. Lueck.

Findings

Finite dimensional classifying spaces exist for certain elementary amenable groups.

Criteria are provided for groups with torsion-free quotients to admit finite dimensional classifying spaces.

Examples show the geometric dimension can be arbitrarily larger than for proper actions.

Abstract

By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality aleph-n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+h+2. We also provide a criterion for groups that fit into an extension with torsion-free quotient to admit a finite dimensional classifying space with virtually cyclic stabilizers. Finally, we exhibit examples of integral linear groups of type F whose geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than the geometric dimension for proper actions. This answers a question posed by W. Lueck.