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Title: Stability, Non-Approximate
d Groups and High-Dimensiona l Expanders
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 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. Using Garland method ( a.k.a. high dimensional expanders as quotients of Bruhat-Tits buildings) , it is shown that some non-residually-
All notions will be explained. Based on joint works with M. De Chiffre, L. Glebsky and A. Thom and with I. Oppenheim.