Expansion, Random Walks and Sieving in SL₂ (mathbb{F}_p [t])

We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of $SL_2 (\mathbb{F}_p [t])$ modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape of such walks from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in $SL_2 (\mathbb{F}_p [t])$.