Stability, cohomology vanishing, and non-approximable groups

Understand

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 $\mathrm{Sym}(n)$ (in the sofic case) or the finite dimensional unitary groups ${\rm U}(n)$ (in the hyperlinear case)? In the case of ${\rm U}(n)$, 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 ${\rm U}(n)$ with respect to the Frobenius norm $\|T\|_{\rm{Frob}}=\sqrt{\sum_{i,j=1}^n|T_{ij}|^2},T=[T_{ij}]_{i,j=1}^n\in\mathrm{M}_n(\mathbb C)$.
• Our 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.