Arithmetic groups

Several well-known open questions, such as: "are all groups sofic or hyperlinear?"  have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric

groups Sym(n) (in the sofic case) or the  unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. 

We answer, for the first time, some of these versions, showing that there exist finitely presented groups which are  not approximated by U(n) with respect to the Frobenius (=L_2) norm and many other norms.

    The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability  and using Garland method , it is shown that 

some non-residually finite groups  (central extensions of some lattices in p-adic Lie groups- a p-adic version of a result of Deligne)  are stable and hence cannot be approximated. 

 

 All notions will be explained.       Based on joint works with M. De Chiffre, L. Glebsky and A. Thom ( arXiv:1711.10238) ) and with I. Oppenheim ( arXiv:1807.06790).