Classical groups, probabilistic methods, and the $(2,3)$-generation problem

Abstract

We study the probability that randomly chosen elements of prescribed type in a finite simple classical group $G$ generate $G$; in particular, we prove a conjecture of Kantor and Lubotzky in this area. The probabilistic approach is then used to determine the finite simple classical quotients of the modular group $\mathrm{PSL}_2(\mathbb{Z})$, up to finitely many exceptions.