Explicit growth and expansion for 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 $\SL_2$.

42 pages; v3, major corrections, including (almost) all numerical constants, and correction of one serious mistake, giving much worse expansion bounds (many thanks to the referee for this paper)