Graphs with large girth and free groups

We use Margulis’ construction together with lattice counting arguments to build Cayley graphs on ${\mathrm{S}\mathrm{L}}_2\left({\mathbb{F}}_p\right),p\to \mathrm{\infty }$ which are d-regular graphs with girth $\ge \frac{2}{3}\frac{\ln \left(n\right)}{\ln (d-1)+\ln \left(C\right)}$ for some absolute constant C.