Expansion, Random Walks and Sieving in $SL_2 (\mathbb{F}_p [t])$

TL;DR

This paper constructs new expander graphs from subgroups of SL_2 over polynomial rings over finite fields, and analyzes random walks on these groups to understand their escape rates from algebraic subsets.

Contribution

It introduces new examples of expander Cayley graphs for groups derived from SL_2 over polynomial rings, with applications to random walk behavior.

Findings

Constructed new expander Cayley graphs from SL_2 over finite field polynomial rings.

Provided bounds on the escape rate of random walks from algebraic subvarieties.

Analyzed the behavior of random walks with respect to squares and reducible polynomials.

Abstract

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