Conjugator Length in Iterated Cyclic Amalgams of Free Groups

Abstract

Let ${\mathcal{A}}_0$ be the set of finitely generated free groups on some countable alphabet and and define ${\mathcal{A}}_r$ inductively as the union of ${\mathcal{A}}_{r-1}$ with the set of amalgamated free products $G_1\ast_{\ZZ}G_2$, where $G_1,G_2\in {\mathcal{A}}_{r-1}$. We show, for all $r$, that every element of ${\mathcal{A}}_r$ has linear conjugator length, quantifying a result of Larsen and extending a result of Kharlampovich and Myasnikov. This is achieved by a linear conjugator length bound on a family of iterated HNN extensions of free groups. Also, in an appendix, we adapt a proof of A. Wei$ß$ to show that all Generalized Baumslag-Solitar Groups have linear conjugator length. As a consequence, we obtain that one-relator groups with non-trivial center have linear conjugator length, giving evidence for a conjecture of Bridson, Riley, and Sale.

arXiv commentIncludes appendix by author