Wednesday, January 31, 2018
01/31/2018 - 4:15pm to 5:15pm
Several well-known open questions (such as: are allgroups sofic or hyperlinear?) have a common form: can all groups beapproximated by asymptotic homomorphisms into the symmetricgroups 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 andnorms. We answer, for the first time, one of these versions,showing that there exist fintely presented groups which arenot approximated by U(n) with respect to the Frobenius (=L2)norm. The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven toimply stability. Combining results of Garland and Deligne , using highdimensional expanders, it is shown that somecentral extensions of some lattices in p-adic Lie groups are Frobenious stableand cannot be Frobenius approximated. All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.