# Graphs with large girth and free groups

## TL;DR

This paper constructs large girth Cayley graphs on SL(2, p) using Margulis' method and lattice counting, achieving graphs with girth proportional to the logarithm of the number of vertices.

## Contribution

It introduces a novel construction of high-girth Cayley graphs on SL(2, p) with explicit girth bounds based on lattice counting techniques.

## Key findings

- Constructed d-regular Cayley graphs with girth ≥ (2/3) * (ln n) / (ln(d-1) + ln C)
- Achieved graphs with girth growing logarithmically with the size of the graph
- Demonstrated the effectiveness of Margulis' construction combined with lattice counting

## Abstract

We use Margulis' construction together with lattice counting arguments to build Cayley graphs on $\mathrm{SL}_{2}\left(\mathbb{F}_{p}\right),\;p\to\infty$ which are d-regular graphs with girth $\geq\frac{2}{3}\frac{\ln\left(n\right)}{\ln\left(d-1\right)+\ln\left(C\right)}$ for some absolute constant C.