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Bounded conciseness in the space of marked groups

Federico Berlai

TL;DR

The paper addresses whether bounded conciseness of words in groups is preserved under limits in the space of marked groups and its relation to residually finite groups. It proves that if a word is boundedly concise in a class \(\mathcal{C}\), then it remains boundedly concise in the class \(\mathcal{L}\) of limits of \(\mathcal{C}\)-groups, using LEF concepts and limit arguments in the marked-group topology. This yields equivalences: a word is boundedly concise in residually finite groups iff it is boundedly concise in LEF groups iff it is concise in LEF groups, providing a reformulation of the Fernández-Alcober–Shumyatsky conjecture in terms of LEF limits. The paper also discusses examples of non-residually finite groups where all words are concise, illustrating the scope and limitations of the bounded conciseness phenomenon and suggesting avenues for tackling the conjecture.

Abstract

We prove that bounded conciseness is a closed property in the space of marked groups. As a consequence, we reformulate a conjecture of Fernández-Alcober and Shumyatsky [7] about conciseness in the class of residually finite groups.

Bounded conciseness in the space of marked groups

TL;DR

The paper addresses whether bounded conciseness of words in groups is preserved under limits in the space of marked groups and its relation to residually finite groups. It proves that if a word is boundedly concise in a class , then it remains boundedly concise in the class of limits of -groups, using LEF concepts and limit arguments in the marked-group topology. This yields equivalences: a word is boundedly concise in residually finite groups iff it is boundedly concise in LEF groups iff it is concise in LEF groups, providing a reformulation of the Fernández-Alcober–Shumyatsky conjecture in terms of LEF limits. The paper also discusses examples of non-residually finite groups where all words are concise, illustrating the scope and limitations of the bounded conciseness phenomenon and suggesting avenues for tackling the conjecture.

Abstract

We prove that bounded conciseness is a closed property in the space of marked groups. As a consequence, we reformulate a conjecture of Fernández-Alcober and Shumyatsky [7] about conciseness in the class of residually finite groups.

Paper Structure

This paper contains 4 sections, 8 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Space of marked groups
  3. Conciseness and bounded conciseness
  4. Some examples

Key Result

Theorem 1

Let $\mathcal{C}$ be a class of groups, $w\in F$ a word in a finitely generated free group, and suppose that the word $w$ is boundedly concise in $\mathcal{C}$. Then the word $w$ is boundedly concise in the class $\mathcal{L}$ of limits of groups in $\mathcal{C}$ in the space of marked groups.

Theorems & Definitions (14)

  • Theorem 1
  • Corollary 2
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 4 more