Graphs with large girth and free groups

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 $\ge \frac{2}{3}\frac{\ln \left(n\right)}{\ln \left(d-1\right)+\ln \left(C\right)}$ for some absolute constant C.

3 figures, 14 pages