On random presentations with fixed relator length

TL;DR

This paper investigates the properties of random groups with fixed relator length as the number of generators grows, revealing phase transitions in triviality, hyperbolicity, and freeness depending on the density parameter.

Contribution

It extends known results by analyzing the fixed relator length model, establishing thresholds for triviality, hyperbolicity, and freeness as the number of generators increases.

Findings

For density > 1/2, the group is trivial or cyclic of order two.

For density < 1/2, the group is infinite and hyperbolic.

For density < 1/k, the group is free, with this threshold being sharp.

Abstract

The standard $(n, k, d)$ model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length $k$ on an $n$-element generating set. Gromov's density model of random groups considers the case where $n$ is fixed, and $k$ tends to infinity. We instead fix $k$, and let $n$ tend to infinity. We prove that for all $k \geq 2$ at density $d > 1/2$ a random group in this model is trivial or cyclic of order two, whilst for $d < \frac{1}{2}$ such a random group is infinite and hyperbolic. In addition we show that for $d<\frac{1}{k}$ such a random group is free, and that this threshold is sharp. These extend known results for the triangular ($k = 3$) and square ($k = 4)$ models of random groups.