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