Stability, non-approximate groups, and high dimensional expanders

Abstract: 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, one of these versions, showing that there exist finitely presented groups which are not 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 to imply stability and using higher dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. All notions will be explained. Joint work with M. De Chiffre, L. Glebsky and A. Thom.

Mathematics

Audience: learners

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