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STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS

Part of: Locally compact groups and their algebras

Published online by Cambridge University Press:  30 March 2020

MARCUS DE CHIFFRE ,
LEV GLEBSKY ,
ALEXANDER LUBOTZKY  and
ANDREAS THOM
MARCUS DE CHIFFRE
Affiliation:
TU Dresden, Germany; [email protected]
LEV GLEBSKY
Affiliation:
L.G., Universidad Autónoma de San Luis Potosí, México; [email protected]
ALEXANDER LUBOTZKY
Affiliation:
A.L., Hebrew University, Israel; [email protected]
ANDREAS THOM
Affiliation:
TU Dresden, Germany; [email protected]

Abstract

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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 $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$, 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 finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$. 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 nonresidually 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.

MSC classification

Primary: 22D10: Unitary representations of locally compact groups
Secondary: 39B82: Stability, separation, extension, and related topics
Type
Topology
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e18
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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