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Word maps and random words

Emmanuel Breuillard, Itay Glazer

TL;DR

This work surveys the landscape of word maps on groups, connecting probabilistic Waring-type questions for finite and algebraic groups with the geometry of representation/character varieties. It highlights how equidistribution over finite fields, via Lang–Weil estimates and deep character bounds, translates into uniform mixing results for word measures on large finite simple groups, and how geometric properties like flatness and generic irreducibility control these phenomena. The authors present a cohesive narrative: (i) LST-type uniform L∞-mixing for nontrivial words on finite simple groups, (ii) representation growth and detailed analysis of the commutator word, (iii) a geometric-analytic proof that the convolution of two word maps has geometrically irreducible generic fibers, and (iv) a thorough treatment of random groups, character varieties, and principal parts of character varieties with effective Chebotarev/Lang–Weil tools. The overall message is that equidistribution, representation theory, and algebraic geometry cooperate to yield uniform global bounds and structural insights across finite and algebraic settings, with concrete consequences for random groups and generic fibers of morphisms.

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.

Word maps and random words

TL;DR

This work surveys the landscape of word maps on groups, connecting probabilistic Waring-type questions for finite and algebraic groups with the geometry of representation/character varieties. It highlights how equidistribution over finite fields, via Lang–Weil estimates and deep character bounds, translates into uniform mixing results for word measures on large finite simple groups, and how geometric properties like flatness and generic irreducibility control these phenomena. The authors present a cohesive narrative: (i) LST-type uniform L∞-mixing for nontrivial words on finite simple groups, (ii) representation growth and detailed analysis of the commutator word, (iii) a geometric-analytic proof that the convolution of two word maps has geometrically irreducible generic fibers, and (iv) a thorough treatment of random groups, character varieties, and principal parts of character varieties with effective Chebotarev/Lang–Weil tools. The overall message is that equidistribution, representation theory, and algebraic geometry cooperate to yield uniform global bounds and structural insights across finite and algebraic settings, with concrete consequences for random groups and generic fibers of morphisms.

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.

Paper Structure

This paper contains 36 sections, 60 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Word maps on finite simple groups: probabilistic results
  4. Motivation
  5. Waring type problems in the setting of word maps
  6. Probabilistic Waring type problems for word maps
  7. Random walk on finite groups
  8. Probabilistic Waring problem: finite simple groups
  9. Commutator word, representation growth, and proof of the probabilistic results
  10. Representation growth
  11. The commutator word and representation growth
  12. The commutator word and representation growth
  13. Proof of Theorem \ref{['thm:LST2']}
  14. Proof of probabilistic result (Theorem \ref{['thm:Larsen Shalev']}) in high rank
  15. Proof of Proposition \ref{['prop:main proposition']}
  16. ...and 21 more sections

Key Result

Theorem 2.4

Let$K$ be an algebraically closed field and let$1\neq w\in F_{r}$. hen for every connected semisimple algebraic $K$-group, the map $w_{\underline{G}}:\underline{G}^{r}\rightarrow\underline{G}$ is dominant. In particular, $\left(w*w\right)_{\underline{G}(K)}:\underline{G}(K)^{2r}\rightarrow\underline

Theorems & Definitions (133)

  • Definition 2.2: GH19GH21
  • Definition 2.3
  • Theorem 2.4: Borel Bor83, and also Larsen Lar04
  • Theorem 2.5: Larsen– Shalev– Tiep, LST11
  • Theorem 2.6: The Ore conjecture 1951, LOST10
  • Definition 2.8
  • Remark 2.9
  • Lemma 2.10
  • proof
  • Definition 2.11
  • ...and 123 more