Explicit growth and expansion for SL_2

We give explicit versions of Helfgott’s Growth Theorem for ${\text{SL}}_2$, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of ${\text{SL}}_2(\text{Zz})$ which are Zariski-dense in ${\text{SL}}_2$.