Several well-known open questions (such as: are all groups sofic/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 finite dimensional 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. This paper answers, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius normOur strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain non-residually finite central extensions of lattices in some simple p-adic Lie groups. These groups act on high rank Bruhat-Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

We will say that the permutations f 1 , ..., f n are an -solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is ≤ . We give a sufficient conditions when near to an -solution exists an exact solution and some examples when there does not exist such a solution.

We give a definition of weakly sofic groups (w-sofic groups). Our definition is a rather natural extension of the definition of sofic groups where instead of the Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non-w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

It has been long conjectured that the crossing number of C m  C n is (m À 2)n, for all m; n such that n ! m ! 3. In this paper, it is shown that if n ! m(m þ 1) and m ! 3, then this conjecture holds. That is, the crossing number of C m  C n is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by