Verbal subgroups of hyperbolic groups have infinite width

TL;DR

This paper proves that in non-elementary hyperbolic groups, the verbal subgroups generated by certain words have infinite width, meaning elements cannot be uniformly expressed as bounded products of word values.

Contribution

It establishes that for a broad class of hyperbolic groups, the verbal subgroups associated with non-trivial words have infinite width, a new result in geometric group theory.

Findings

Verbal subgroups in hyperbolic groups have infinite width.

No finite bound exists for expressing elements as products of word values.

The result applies to non-elementary hyperbolic groups with specific word conditions.

Abstract

Let $G$ be a non-elementary hyperbolic group. Let $w$ be a group word such that the set $w[G]$ of all its values in $G$ does not coincide with $G$ or 1. We show that the width of verbal subgroup $w(G)=<w[G]>$ is infinite. That is, there is no such $l\in\mathbb Z$ that any $g\in w(G)$ can be represented as a product of $\le l$ values of $w$ and their inverses.