Explicit growth and expansion for SL_2

TL;DR

This paper provides explicit versions of key theorems related to growth and expansion in the special linear group SL_2, enhancing understanding of subgroup behavior and Cayley graph expansion.

Contribution

It offers explicit formulations of Helfgott's Growth Theorem and the Bourgain-Gamburd expansion argument for Zariski-dense subgroups of SL_2.

Findings

Explicit growth bounds for subgroups of SL_2

Enhanced expansion properties of Cayley graphs modulo primes

Improved understanding of subgroup dynamics in SL_2

Abstract

We give explicit versions of Helfgott's Growth Theorem for $\SL_2$, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of $\SL_2(\Zz)$ which are Zariski-dense in $\SL_2$.