  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
$SL_2 (\mathbb{F}_p [t])$.
