Generic thinness in finitely generated subgroups of ${\text{SL}}_n(\mathbb{Z})$

Abstract

We show that for any $n\ge 2$, two elements selected uniformly at random from a \emph{symmetrized} Euclidean ball of radius $X$ in ${\text{SL}}_n(\mathbb{Z})$ will generate a thin free group with probability tending to $1$ as $X\to \infty .$ This is done by showing that the two elements will form a ping-pong pair, when acting on a suitable space, with probability tending to $1$. On the other hand, we give an upper bound less than $1$ for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where $n>2$.