Verbal subgroups of hyperbolic groups have infinite width

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.

To appear in Journal of the London Mathematical Society. 22 pages, 8 figures

References in corpus (2)

• Foliations for solving equations in groups: free, virtually free, and hyperbolic groups
• On Rationality of Verbal Subsets In a Group

Cited by in corpus (2)

• Words have bounded width in $SL(n,\mathbb{Z})$
• The verbal width of acylindrically hyperbolic groups is infinite