# Girth, words and diameter

## TL;DR

This paper investigates the girth of Cayley graphs of finite classical and symmetric groups, providing bounds on word probabilities and exploring relationships between girth and diameter in random Cayley graphs.

## Contribution

It introduces a nearly optimal bound on the probability that a word evaluates to identity in finite classical groups, with applications to girth and diameter analysis.

## Key findings

- Bound on probability that a word evaluates to 1 in G
- Analysis of girth in random directed Cayley graphs
- Relationship between girth and diameter in finite simple groups

## Abstract

We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.