Word maps and random words

Emmanuel Breuillard; Itay Glazer

Home/Archive/arXiv

arXiv · 2502.00417

Word maps and random words

Emmanuel Breuillard·Itay Glazer

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GR math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Emmanuel Breuillard, Itay Glazer. 2025-02-01. Word maps and random words. https://arxiv.org/abs/2502.00417

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On commensurators of free groups and free pro-p groups

We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.

math.GR↗

arXiv2026

Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry

We study the two $\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\ell^{2}$-Dirichlet subspaces of $\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\mathcal{D}_{2}\left(G,λ\right)=\mathcal{D}_{2}\left(G,ρ\right)\quad\Longleftrightarrow\quad G \text{ is an FC-group}.$$ The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.

math.GR↗

arXiv2026

Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces

We establish an unconditional analogue of Sylow's first theorem for finite skew braces, and deduce an analogue of Cauchy's theorem. We also prove an analogue of the existence part of Hall's theorem for finite skew braces with soluble additive and multiplicative groups. We make some observations regarding the number of Sylow subskew braces of a skew brace in various cases. By applying these results we streamline the classification of skew braces of order $ pq $, where $ p,q $ are distinct prime numbers.

math.GR↗