Arithmetic Groups

In the first part of this talk, we take the ideas of the second talk and focus on the case of (arithmetic) lattices in $PSL(2,R)$ and $PSL(2,C)$. The required representation rigidity is achieved by what we call Galois rigidity. In particular if $\Gamma_1$ is a Galois rigid lattice in $PSL(2,R)$ or $PSL(2,C)$ and $\Gamma_2$ a finitely generated residually finite group with $\widehat{\Gamma}_1\cong \widehat{\Gamma}_2$, then we first show how to build a $PSL(2,R)$ or $PSL(2,C)$ Zariski dense representation of $\Gamma_2$ using local representations.

Under additional algebraic assumptions on $\Gamma_1$, we will discuss how to further refine this to produce a homomorphism from $\Gamma_2$ to $\Gamma_1$.

In the second part of the talk, we will next proceed to describe explicit examples of arithmetic Kleinian groups where we can execute the set up described above, and where we can also execute the final part of the program to prove that these groups are profinitely rigid (as advertised in Lecture 1).

Time permitting we finish by commenting on more recent work: (1) that certain closed Seifert fibered spaces have fundamental groups that are profinitely rigid, and (2) using these examples to build examples of groups that are profinitely rigid amongst finitely presented groups but not finitely generated groups.