Non Hyperbolic Free-By-Cyclic and One-Relator Groups

TL;DR

This paper demonstrates that certain free-by-cyclic groups act properly cocompactly on CAT(0) square complexes and classifies groups defined by 2-generator 1-relator presentations, revealing their structural properties and limitations.

Contribution

It establishes the proper cocompact action of F(2)-by-Z groups on CAT(0) square complexes and classifies 2-generator 1-relator groups as either SQ-universal, cyclic, or isomorphic to BS(1,j).

Findings

F(2)-by-Z groups act on CAT(0) square complexes.

All known 2-generator 1-relator groups are either SQ-universal, cyclic, or BS(1,j).

Some free-by-cyclic groups are not relatively hyperbolic.

Abstract

We show that the free-by-cyclic groups of the form F(2)-by-Z act properly cocompactly on CAT(0) square complexes. We also show using generalised Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to BS(1,j). Finally we consider free-by-cyclic groups which are not relatively hyperbolic with respect to any collection of subgroups.