Word maps and random words

TL;DR

This paper explores recent advances in the study of word maps on algebraic and finite simple groups, focusing on their mixing properties, fiber geometry, and applications to character varieties and irreducibility proofs.

Contribution

It provides new insights into the geometry of fibers of word maps, discusses recent character bounds, and offers a novel proof of a theorem on the irreducibility of generic fibers.

Findings

Analysis of mixing properties of word maps

Results on the geometry of fibers in algebraic groups

A new proof of Hrushovski's theorem on irreducibility

Abstract

We discuss some recent results by a number of authors regarding word maps on algebraic groups and finite simple groups, their mixing properties and the geometry of their fibers, emphasizing the role played by equidistribution results in finite fields via recent advances on character bounds and non-abelian arithmetic combinatorics. In particular, we discuss character varieties of random groups. In the last section, we give a new proof of a recent theorem of Hrushovski about the geometric irreducibility of the generic fibers of convolutions of dominant morphisms to simply connected algebraic groups. These notes stem out of lectures given by the authors in Oxford, and by the first author in ICTS Bangalore, in spring 2024.