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.

中文翻译：

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稳定性、上同调消失和不可逼近群

几个众所周知的开放性问题（例如：所有群都是sofic/hyperlinear？）有一个共同的形式：所有群都可以通过渐近同态近似为对称群 $\text{符号}(n)$ （在 sofic 的情况下）或有限维酉群 $\文本{U}(n)$ （在超线性的情况下）？如果是 $\文本{U}(n)$ ，可以针对不同的指标和规范提出问题。本文首次回答了其中一个版本，表明存在有限呈现的群，它们是不是近似于 $\文本{U}(n)$ 关于 Frobenius 范数 $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij }]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$ . 我们的策略是证明一些高维上同调消失现象意味着稳定，也就是说，每个有限维酉群的 Frobenius 近似同态都接近于实际的同态。这与一些简单的格子的某些非剩余有限中心扩展的存在结果相结合 $p$ -adic 李群。这些群作用于高级 Bruhat-Tits 建筑物并满足所需的消失上同调现象，因此是稳定的而不是 Frobenius 近似的。