Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

TL;DR

This paper investigates a class of cyclically presented groups, demonstrating their coherence, subgroup separability, and structural properties, including classifications of finite subgroups and implications for related conjectures.

Contribution

It introduces and analyzes the class al M cyclically presented groups, establishing their key algebraic and geometric properties and classifying their finite subgroups.

Findings

Groups in al M are coherent and subgroup separable.

Finite subgroups are all metacyclic.

Many groups are virtually free or have geometric dimension two.

Abstract

We study a class $\mathfrak{M}$ of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class $\mathfrak{M}$ are are coherent, subgroup separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all metacyclic. Many of the groups in $\mathfrak{M}$ are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in $\mathfrak{M}$, giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class $\mathfrak{M}$ that has fixed point free shift automorphism must be cyclic.