Character estimates for finite simple groups and applications

Abstract

Let $G$ be a finite simple group, $\chi$ an irreducible complex character, and $g$ an element of $G$. It is often desirable to have upper bounds for $|\chi (g)|$ in terms of $\chi (1)$ and some measure of the regularity of $g$. This paper reviews what is known in this direction and presents typical applications of such bounds: to proving certain products of conjugacy classes cover $G$, to solving word equations over $G$, and to counting homomorphisms from a Fuchsian group to $G$.