Classifying Spaces with Virtually Cyclic Stabilisers for Certain Infinite Cyclic Extensions

TL;DR

This paper constructs models for the classifying space of certain infinite cyclic extensions with free actions on conjugacy classes, providing minimal dimension bounds and explicit 2D models for specific Baumslag-Solitar groups.

Contribution

It introduces a method to construct models for Evc(G) for a class of infinite cyclic extensions and establishes minimal dimension bounds, including explicit 2D models for Baumslag-Solitar groups.

Findings

Constructed a model for Evc(G) for certain infinite cyclic extensions.

Provided bounds for the minimal dimension of Evc(G).

Developed a 2-dimensional minimal model for Baumslag-Solitar groups BS(1,m).

Abstract

Let G be an infinite cyclic extension, 1 -> B -> G -> Z -> 1, of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain ascending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|> 1. We construct a model for Evc(G), the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimum dimension of Evc(G). We construct a 2-dimensional model for Evc(G) where G is a soluble Baumslag-Solitar BS(1,m) group with |m|>1 and we show that this model for Evc(G) is of minimal dimension.