First-order rigidity, bi-interpretability, and congruence subgroups
I'll describe a method for analyzing the first-order theory of an arithmetic group using its congruence quotients. When this method works, it gives a strong form of first-order rigidity together with a complete description of the collection of definable subsets in the group. I'll mention two cases where this method works: one is for uniform lattices in orthogonal groups (which are, therefore, first-order rigid); the other is for families of arithmetic groups. One corollary is that there is a first-order statement that holds in a finitely generated group if and only if this group is isomorphic to a group of the form SL(n,A), where n is at least 3 and A is a ring of integers in a number field. This is joint work with Chen Meiri.