Counting embeddings of free groups into ${\mathrm{SL}}_2(\mathbb{Z})$ and its subgroups

Abstract

We show that if one selects uniformly independently and identically distributed matrices $A_1,\dots ,A_s\in {\mathrm{SL}}_2(\mathbb{Z})$ from a ball of large radius $X$ then with probability at least $1-X^{-1+o(1)}$ the matrices $A_1,\dots ,A_s$ are free generators for a free subgroup of ${\mathrm{SL}}_2(\mathbb{Z})$. Furthermore, to show the flexibility of our method we do similar counting for matrices from the congruence subgroup ${\Gamma }_0(Q)$ uniformly with respect to the positive integer $Q\le X$. This improves and generalises a result of E. Fuchs and I. Rivin (2017) which claims that the probability is $1+o(1)$. We also disprove one of the statements in their work that has been used to deduce their claim.