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


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 $\mathrm{SL}_2(\mathbb{F}_p)$.


Jean Bourgain

School of Mathematics
Institute for Advanced Study
Princeton, NJ 08540
United States

Alex Gamburd

Department of Mathematics
University of California at Santa Cruz
Santa Cruz, CA 95064
United States