Generic Thinness in Finitely Generated Subgroups of SL$_n(\mathbb Z)$

Abstract

We show that for any |$n\geq 2$|⁠, two elements selected uniformly at random from a symmetrized Euclidean ball of radius |$X$| in |$\text{SL}_n(\mathbb Z)$| will generate a thin free group with probability tending to |$1$| as |$X\rightarrow \infty.$| 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 |$<1$| for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where |$n>2$|⁠.

Locations

• International Mathematics Research Notices - View - PDF