Outer commutator words are uniformly concise

TL;DR

This paper proves that outer commutator words are uniformly concise, establishing bounds on the verbal subgroup's order based solely on the number of distinct values, using a novel tree-based representation and structure theorem.

Contribution

It introduces a new binary tree representation of outer commutator words and proves a structure theorem for their verbal subgroups, leading to uniform conciseness results.

Findings

Outer commutator words are uniformly concise.

The order of the verbal subgroup is bounded by a function of the number of values.

A new tree-based representation aids in structural analysis.

Abstract

We prove that outer commutator words are uniformly concise, i.e. if an outer commutator word w takes m different values in a group G, then the order of the verbal subgroup w(G) is bounded by a function depending only on m and not on w or G. This is obtained as a consequence of a structure theorem for the subgroup w(G), which is valid if G is soluble, and without assuming that w takes finitely many values in G. More precisely, there is an abelian series of w(G), such that every section of the series can be generated by values of w all of whose powers are also values of w in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction.