On the measure of definable sets in profinite groups

Martina Conte; Matteo Vannacci

Home/Archive/arXiv

arXiv · 2609.25448

On the measure of definable sets in profinite groups

Martina Conte·Matteo Vannacci

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the (Haar) measure of verbal sets in profinite groups, and especially in free profinite groups. Relating the measure of a verbal set with the width of the corresponding word, we find verbal sets of positive measure different from one. We prove a sufficient condition for the measure of a verbal set to be zero in a free profinite group of finite rank, from which we deduce that certain existentially definable sets in free profinite groups of finite rank can only have measure zero or one. Finally, we compute the measure of definable sets in torsion-free finitely generated abelian groups.

Explore related subjects

math.GR math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Martina Conte, Matteo Vannacci. 2026-09-21. On the measure of definable sets in profinite groups. https://arxiv.org/abs/2609.25448

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

KEEP EXPLORING

Related papers

arXiv2026

Margulis-Soifer theorem for one-relator groups

We establish the Margulis-Soifer dichotomy for one-relator groups: every one-relator group is either virtually solvable or has a maximal subgroup of infinite index. We also present examples of one-relator groups with and without free maximal subgroups of infinite index, as well as examples that possess both free and non-free infinite index maximal subgroups. Triviality of the Frattini subgroup is also shown for all non-solvable one-relator groups. We close the paper with a short list of questions.

math.GR↗

arXiv2026

Finite quotients of spherical Artin groups

We show the smallest non-abelian quotients of spherical and affine Artin groups are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. We deduce irreducible spherical Artin groups are determined by their finite quotient groups.

math.GR↗

arXiv2026

Cosets with constant characteristic polynomial

Let H be a linear group. We show that if there is an invertible matrix x such that all the elements of xH share the same characteristic polynomial then H is virtually solvable. There are plenty of applications that will be presented in future paper. Here, we discuss some applications to the generalized Weigold conjecture and present an alternative straightforward proof of the Formanek--Procesi nonlinearity theorem for Aut(F_n), n>2, over every field. When n>5 our non-linearity proof gives a stronger result than the original Formanek--Procesi theorem.

math.GR↗