Arithmetic Groups

If a monomorphism of abstract groups $H\hookrightarrow G$ induces an isomorphism of profinite completions, then $(G, H)$ is called a Grothendieck pair, recalling the fact that Grothendieck asked about the existence of such pairs with $G$ and $H$ finitely presented and residually finite. A finitely generated group is said to profinitely rigid if it can be distinguished from all other finitely generated, residually finite groups by its profinite completion. There is a rich history around the study of these ideas, which I shall sketch before discussing the recent surge of activity in this area. The recent progress draws on advances in geometric group theory and low dimensional topology; arithmetic orbifolds are the objects of primary interest. Among these arithmetic orbifolds, we find the first examples of full-sized groups that are profinitely rigid.