Random walks on direct products of groups

Abstract

Suppose ${\mathrm{SL}}_2({\mathbb{F}}_p)\times {\mathrm{SL}}_2({\mathbb{F}}_p)$ is generated by a symmetric set $S$ of cardinality $n$ where $p$ is a prime number. Suppose the Cheeger constants of the Cayley graphs of ${\mathrm{SL}}_2({\mathbb{F}}_p)$ with respect to ${\pi }_L(S)$ and ${\pi }_R(S)$ are at least $c_0$, where ${\pi }_L$ and ${\pi }_R$ are the projections to the left and the right components of ${\mathrm{SL}}_2({\mathbb{F}}_p)\times {\mathrm{SL}}_2({\mathbb{F}}_p)$, respectively. Then the Cheeger constant of the Cayley graph of ${\mathrm{SL}}_2({\mathbb{F}}_p)\times {\mathrm{SL}}_2({\mathbb{F}}_p)$ with respect to $S$ is at least $c$ where $c$ is a positive number which only depends on $n$ and $c_0$. This gives an affirmative answer to a question of Lindenstrauss and Varjú.