Uniform expansion bounds for Cayley graphs of $\mathrm{SL}_2(F_p)$

Abstract

We prove that Cayley graphs of $\mathrm{SL}_2(\mathbb{F}_p)$ are expanders with respect to the projection of any fixed elements in $\mathrm{SL}(2, \mathbb{Z})$ generating a non-elementary subgroup, and with respect to generators chosen at random in $\mathrm{SL}_2(\mathbb{F}_p)$.