Parallel geodesics and minimal stable length of random groups

TL;DR

This paper investigates the structure of random groups at low density, showing that long parallel geodesics are highly constrained and establishing that the minimal stable length in such groups is exactly one.

Contribution

It introduces a new geometric property of random groups at density less than 1/6, providing bounds on parallel geodesics and determining the minimal stable length.

Findings

Long enough parallel geodesics are contained in a single-layer van Kampen diagram.

Number of pairwise parallel geodesics is bounded depending only on the density.

Minimal stable length of the group is exactly 1 at density less than 1/6.

Abstract

We show that for any pair of long enough parallel geodesics in a random group $G_\ell(m,d)$ with $m$ generators at density $d<1/6$, there is a van Kampen diagram having only one layer of faces. Using this result, we give an upper bound, depending only on $d$, of the number of pairwise parallel geodesics in $G_\ell(m,d)$ when $d<1/6$. As an application, we show that the minimal stable length of $G_\ell(m,d)$ at $d<1/6$ is exactly $1$.