Papers for

computational group theorists

Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.

Linear bounds on conjugator length in iterated free group constructions

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 $G_1*_{\mathbb{Z}}G_2$, 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. 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.

Thu 17 SeptDiscrete Mathematics

The gist

Certain groups built step-by-step from simple building blocks, called free groups, can have tricky elements that are related by 'conjugation.' This paper shows that the length needed to describe these relations grows only in a simple, predictable way (linearly) as the groups get more complex by joining along circles. The authors extend previous results to a broader family of groups and provide new proofs that reinforce an existing conjecture about one-relator groups. Their work clarifies how complicated it is to find these conjugations as group constructions become more layered.

Open → 2609.21116v1