On the uniform spread of almost simple symplectic and orthogonal groups

TL;DR

This paper proves a stronger form of a conjecture regarding the generation properties of almost simple symplectic and orthogonal groups, providing bounds and asymptotics for their uniform spread.

Contribution

It extends the conjecture's proof to symplectic and orthogonal groups, offering new bounds and asymptotic results for their uniform spread.

Findings

Proved the conjecture for almost simple symplectic groups.

Established lower bounds for the uniform spread of these groups.

Derived asymptotic behavior related to group generation properties.

Abstract

A group is $\frac{3}{2}$-generated if every non-identity element is contained in a generating pair. A conjecture of Breuer, Guralnick and Kantor from 2008 asserts that a finite group is $\frac{3}{2}$-generated if and only if every proper quotient of the group is cyclic, and recent work of Guralnick reduces this conjecture to almost simple groups. In this paper, we prove a stronger form of the conjecture for almost simple symplectic and odd-dimensional orthogonal groups. More generally, we study the uniform spread of these groups, obtaining lower bounds and related asymptotics. This builds on earlier work of Burness and Guest, who established the conjecture for almost simple linear groups.