On conciseness of words in profinite groups

TL;DR

This paper confirms a conjecture relating the countability of word values in profinite groups to the finiteness of the corresponding verbal subgroup, specifically for multilinear commutator words, squares, and certain commutators.

Contribution

It proves the conjecture for specific classes of words, advancing understanding of the structure of profinite groups based on word value properties.

Findings

Confirmed the conjecture for multilinear commutator words

Confirmed the conjecture for the word x^2

Confirmed the conjecture for the word [x^2,y]

Abstract

Let w be a group word. It is conjectured that if w has only countably many values in a profinite group G, then the verbal subgroup w(G) is finite. In the present paper we confirm the conjecture in the cases where w is a multilinear commutator word, or the word x^2, or the word [x^2,y].