Perfect Prishchepov groups

TL;DR

This paper classifies perfect cyclically presented groups, especially Prishchepov and Fibonacci-type groups, based on their defining parameters, confirming a conjecture and providing a comprehensive understanding of their structure.

Contribution

It offers a classification of perfect Prishchepov groups and Fibonacci-type groups, confirming a conjecture and linking these groups to Labelled Oriented Graph groups.

Findings

Classification of perfect Prishchepov groups $P(r,n,k,s,q)$

Complete classification of perfect Fibonacci-type groups $H(r,n,s)$

Identification of which $H(r,n,s)$ are connected Labelled Oriented Graph groups

Abstract

We study cyclically presented groups of type $\mathfrak{F}$ to determine when they are perfect. It turns out that to do so, it is enough to consider the Prishchepov groups, so modulo a certain conjecture, we classify the perfect Prishchepov groups $P(r,n,k,s,q)$ in terms of the defining integer parameters $r,n,k,s,q$. In particular, we obtain a classification of the perfect Campbell and Robertson's Fibonacci-type groups $H(r,n,s)$, thereby proving a conjecture of Williams, and yielding a complete classification of the groups $H(r,n,s)$ that are connected Labelled Oriented Graph groups.