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

TL;DR

This paper demonstrates that randomly chosen matrices in $ ext{SL}_2( ext{Z})$ and its subgroups are likely to generate free subgroups, improving previous probabilistic results and correcting earlier claims.

Contribution

It provides new probabilistic estimates for free subgroup generation in $ ext{SL}_2( ext{Z})$ and its congruence subgroups, extending and refining prior work.

Findings

High probability of free subgroup generation from random matrices

Extension of counting methods to congruence subgroups $ ext{Gamma}_0(Q)$

Disproof of a previous statement used in earlier proofs

Abstract

We show that if one selects uniformly independently and identically distributed matrices $A_1, \ldots, A_s \in \mathrm{SL}_2(\mathbb{Z})$ from a ball of large radius $X$ then with probability at least $1 - X^{-1 + o(1)}$ the matrices $A_1, \ldots, A_s$ 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.