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Verbal subgroups of hyperbolic groups have infinite width. (English) Zbl 1339.20036

Summary: Let \(G\) be a non-elementary hyperbolic group. Let \(w\) be a proper group word. We show that the width of the verbal subgroup \(w(G)=\langle w[G]\rangle\) is infinite. That is, there is no \(l\in\mathbb Z\) such that any \(g\in w(G)\) can be represented as a product of at most \(l\) values of \(w\) and their inverses. As a consequence, we obtain the same result for a wide class of relatively hyperbolic groups.

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
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