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On profinite groups admitting a word with only few values

Pavel Shumyatsky

Abstract

A group-word $w$ is called concise if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G$. It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word $w$ is strongly concise in profinite groups, that is, $w(G)$ is finite whenever $G$ is a profinite group in which $w$ takes less than $2^{\aleph_0}$ values. In this paper we show that if the word $w$ takes less than $2^{\aleph_0}$ values in a profinite group $G$ then $w(w(G))$ is finite.

On profinite groups admitting a word with only few values

Abstract

A group-word is called concise if the verbal subgroup is finite whenever takes only finitely many values in a group . It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word is strongly concise in profinite groups, that is, is finite whenever is a profinite group in which takes less than values. In this paper we show that if the word takes less than values in a profinite group then is finite.
Paper Structure (3 sections, 4 equations)

This paper contains 3 sections, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main theorem

Theorems & Definitions (2)

  • proof
  • proof