Virtually free-by-cyclic groups

TL;DR

This paper characterizes virtually free-by-cyclic groups within hyperbolic, virtually compact special groups using homological methods and proves that all one-relator groups with torsion are virtually free-by-cyclic, confirming Baumslag's conjecture.

Contribution

It provides a homological characterization of virtually free-by-cyclic groups in a specific class and proves a longstanding conjecture about one-relator groups with torsion.

Findings

Homological characterization of virtually free-by-cyclic groups.

All one-relator groups with torsion are virtually free-by-cyclic.

Many known coherent groups are shown to be virtually free-by-cyclic.

Abstract

We obtain a homological characterisation of virtually free-by-cyclic groups among groups that are hyperbolic and virtually compact special. As a consequence, we show that many groups known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, we show that all one-relator groups with torsion are virtually free-by-cyclic, solving a conjecture of Baumslag.