Zero-one generation laws for finite simple groups

TL;DR

This paper proves a zero-one law for the probability that finite simple groups of Lie type are generated by random elements from certain subvarieties, with applications to random generation of these groups by elements of orders 2 and 3.

Contribution

It establishes a zero-one law for generation probabilities in finite simple groups of Lie type and applies it to show generic (2,3)-generation for most such groups.

Findings

Probability tends to 1 or 0 as q increases

Finite simple groups are generically (2,3)-generated

Results extend to varying characteristic cases

Abstract

Let $G$ be a simple algebraic group over the algebraic closure of $GF(p)$ ($p$ prime), and let $G(q)$ denote a corresponding finite group of Lie type over $GF(q)$, where $q$ is a power of $p$. Let $X$ be an irreducible subvariety of $G^r$ for some $r\ge 2$. We prove a zero-one law for the probability that $G(q)$ is generated by a random $r$-tuple in $X(q) = X\cap G(q)^r$: the limit of this probability as $q$ increases (through values of $q$ for which $X$ is stable under the Frobenius morphism defining $G(q)$) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs $G(q)$ to be generated by an $r$-tuple in $X(q)$ for two sufficiently large values of $q$. We also prove a version of this result where the underlying characteristic is allowed to vary. In our main application, we apply these results to the case where $r=2$ and the irreducible subvariety $X = C\times D$, a product of two conjugacy classes of elements of finite order in $G$. This leads to new results on random $(2,3)$-generation of finite simple groups $G(q)$ of exceptional Lie type: provided $G(q)$ is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate $G(q)$ tends to $1$ as $q \rightarrow \infty$. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and $PSp_4(q)$) are randomly $(2,3)$-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie type.