Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups

TL;DR

This paper classifies non-elementary hyperbolic groups of Fibonacci type and T(5) cyclically presented groups, providing almost complete results and highlighting unresolved cases, with implications for the Tits alternative.

Contribution

It offers a near-complete classification of hyperbolic groups in these classes, identifying specific cases and conditions affecting hyperbolicity.

Findings

Most Fibonacci type groups are hyperbolic, except H(9,4) and H(9,7).

If H(9,4) is torsion-free, it is not hyperbolic.

T(5) cyclically presented groups satisfy the Tits alternative.

Abstract

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9,4), H(9,7). We show that if H(9,4) is torsion-free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds.