Stability, cohomology vanishing, and non-approximable groups

TL;DR

This paper demonstrates the existence of finitely presented groups that cannot be approximated by finite-dimensional unitary groups under the Frobenius norm, linking cohomology vanishing phenomena to stability and non-approximability.

Contribution

It provides the first example of groups not Frobenius-approximable, connecting cohomology vanishing with stability and approximation properties.

Findings

Existence of non-approximable finitely presented groups.

Cohomology vanishing implies stability of approximate homomorphisms.

Certain non-residually finite central extensions are stable and non-approximable.

Abstract

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. These groups act on high rank Bruhat-Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.