Abels's groups revisited

TL;DR

This paper generalizes Abels's groups to construct examples of virtually torsion-free groups with specific Bredon and classical finiteness lengths, demonstrating geometric methods for property verification.

Contribution

It introduces a generalized class of groups with prescribed finiteness properties, expanding understanding of Bredon and classical finiteness lengths.

Findings

Constructed groups with Bredon-finiteness length m-1 and classical finiteness length n-1

Demonstrated geometric verification methods for Bredon-finiteness properties

Extended the class of known examples of virtually torsion-free groups

Abstract

We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown's criterion due to Martin Fluch and the author.