Arithmetic Groups

Taking up the terminology established in the first lecture, in 1970 Grothendieck showed that when two groups $(G,H)$ form a Grothendieck pair, there is an equivalence of their linear representations. For recent work showing that certain groups are profinitely rigid, one step is to relate linear representations of groups which are only profinitely equivalent, $\widehat{G} \cong \widehat{H}$, when $G$ has certain representation rigidity properties. In particular, one can produce a representation of $H$ with arithmetic information related to that of $G$. In some cases, this allows one to reduce the question of absolute profinite rigidity, $G$ is profinitely rigid among all finitely generated residually finite groups, to a relative profinite rigidity question, that $G$ is profinitely rigid within some restricted class.

In this lecture, I will discuss this relationship of linear representations in detail to convey its reach and its limitations. I will explain the specific consequences which follow when considering higher-rank arithmetic groups, and the role of the congruence subgroup property in these cases. I will also give some indication of how the proofs come from relating the $p$-adic representations of a group with those of its profinite completion.