Primitivity rank, one-relator groups and hyperbolicity

Abstract: The primitivity rank of an element \(w\) of a free group \(F\) is defined as the minimal rank of a subgroup containing \(w\) as an imprimitive element. Recent work of Louder and Wilton has shown that there is a striking connection between this quantity and the subgroup structure of the one-relator group \(F/\langle\langle w\rangle\rangle\). In this talk, I will start by motivating the study of one-relator groups and survey some recent advancements. Then, I will show that one-relator groups whose defining relation has primitivity rank at least 3 are hyperbolic, confirming a conjecture of Louder and Wilton. Finally, I will discuss the ingredients that go into proving this result.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar