** "Geometry and Analysis on Groups" Research Seminar **

**Title:**
Subgroup rigidity in \(\mathrm{Out}(F_n)\).

**Speaker:**
Ric Wade (Oxford)

**Abstract:**
A subgroup \(H\) of a group \(G\) is rigid if every injective map from \(H\) into \(G\) is induced by conjugation by an element of \(G\). In some recent work with Hensel and Horbez, we show that many subgroups of \(\mathrm{Out}(F_n)\) are rigid. I will talk about some motivation for this coming from abstract commensurators of groups, particularly previous work by Ivanov on mapping class groups. The main techniques involve combining combinatorial rigidity of a certain \(\mathrm{Out}(F_n)\)-graph with structural results about its vertex stabilizers. We shall see that a key tool for the second part involves understanding how direct products of groups can act on Gromov hyperbolic spaces.