Commensurators and classifying spaces with virtually cyclic stabilizers

TL;DR

This paper investigates the structure of commensurators of virtually cyclic groups and constructs finite-dimensional classifying spaces with virtually cyclic stabilizers for certain classes of groups, advancing understanding in geometric group theory.

Contribution

It introduces new methods to build finite-dimensional classifying spaces with virtually cyclic stabilizers for locally finite-by-virtually cyclic and elementary amenable groups.

Findings

Existence of finite-dimensional classifying spaces for groups of cardinality aleph_n

Dimension bounds for these classifying spaces based on group properties

Extension of results to elementary amenable groups with finite Hirsch length

Abstract

By examining commensurators of virtually cyclic groups, we show that for each natural number n, any locally finite-by-virtually cyclic group of cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+3. As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers.