Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture

TL;DR

This paper establishes bounds on elements in finite classical groups that guarantee they can be expressed as products of conjugates of certain regular semisimple elements, advancing understanding of element factorizations and asymptotic behavior related to Thompson's conjecture.

Contribution

It provides new bounds for expressing elements as products of conjugates of regular semisimple elements in finite classical groups, extending results to orthogonal and symplectic groups and analyzing asymptotic cases.

Findings

Bound B(k) for expressing elements as products of conjugates of g

Results for classical groups over finite fields, including orthogonal and symplectic cases

Asymptotic results for prime n=p with specific element orders

Abstract

For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of $\le k$ pairwise distinct monic irreducible polynomials over $\mathbb{F}_q$, then every element $x$ of $\operatorname{SL}_n(q)$ of support at least $B$ is the product of two conjugates of $g$. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions $(p,q)$, in the special case that $n=p$ is prime, if $g$ has order $\frac{q^p-1}{q-1}$, then every non-scalar element $x \in \operatorname{SL}_p(q)$ is the product of two conjugates of $g$. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.