Cyclically presented groups with length four positive relators

TL;DR

This paper classifies when cyclically presented groups with length four positive relators are finite or aspherical, linking these properties to the dynamics of a shift action on the group.

Contribution

It provides a classification of finiteness and mostly asphericity for these groups, and relates these properties to shift dynamics, advancing understanding of their structure.

Findings

Finiteness of the groups is classified.

Asphericity is classified in most cases.

The fixed point subgroup of the shift is always finite.

Abstract

For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying presentations. We show that the fixed point subgroup of the shift $x_i \mapsto x_{i+1}$ is always finite and we relate finiteness of $G$ and asphericity to the dynamics of the shift action by the cyclic group of order $n$ on the nonidentity elements of $G$.