Conjugator Length in Iterated Cyclic Amalgams of Free Groups

Abstract: Let $\mathcal{A}<em>0$ be the set of finitely generated free groups on some countable alphabet and and define $\mathcal{A}_r$ inductively as the union of $\mathcal{A}</em>{r-1}$ with the set of amalgamated free products $G_1*<em>{\mathbb{Z}}G_2$, where $G_1,G_2\in \mathcal{A}</em>{r-1}$. 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. Weiß to show that all Generalized Baumslag-Solitar Groups have linear conjugator length. As a consequence, we obtain that one-relator groups with non-trivial have linear conjugator length, giving evidence for a conjecture of Bridson, Riley, and Sale.