arXiv 1809.006322018-09math.GR

群稳定性和性质（T）

Group stability and Property (T)

Oren Becker, Alexander Lubotzky

备注：: 20 pages; v2 includes new results in Section 3 and an appendix

摘要：

近年来，人们对有限生成群$Γ$相对于一系列群$\left\{G_{n}\right\}_{n=1}^{\infty}$的稳定性产生了相当大的兴趣，这些群配备有双不变度量$\left\{d_{n}\right\}_{n=1}^{\infty}$。我们考虑$G_{n}=\operatorname{U}\left(n\right)$（resp. $G_{n}=\operatorname{Sym}\left(n\right)$），配备有归一化的希尔伯特-施密特度量$d_{n}^{\operatorname{HS}}$（分别为归一化汉明度量$d_{n}^{\operatorname{Hamming}}$）。我们的主要结果是，如果$Γ$是无限的，超线性（分别）。sofic）并且具有性质$\operatorname{(T)}$，则其相对于$\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$（分别地，$\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$）。这回答了Hadwin和Shulman关于$\operatorname{SL}_{3}\left(\mathbb{Z}\right)$稳定性的问题。我们还推导出映射类组$\operatorname{MCG}\left(g\right)$，$g\geq 3$和$\operatorname{Aut}\left(\mathbb{F}_n\right)$，$n\geq 3$相对于$\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$是不稳定的。我们的主要结果显示了$\operatorname{U}\left(n\right)$上的归一化Hilbert-Schmidt度量和（非归一化）$p$-Schatten度量的稳定性之间的差异，因为许多具有$\operatorname{(T)}$性质的群相对于后者度量是稳定的，如De Chiffre-Glebsky-Lubotzky-Thom和Lubotzky-Oppenheim所示。我们提出了一个更灵活的稳定性概念，可以修复$\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$和$\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$稳定性的不足。

英文摘要：

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group $Γ$ with respect to a sequence of groups $\left\{G_{n}\right\}_{n=1}^{\infty}$, equipped with bi-invariant metrics $\left\{d_{n}\right\}_{n=1}^{\infty}$. We consider the case $G_{n}=\operatorname{U}\left(n\right)$ (resp. $G_{n}=\operatorname{Sym}\left(n\right)$), equipped with the normalized Hilbert-Schmidt metric $d_{n}^{\operatorname{HS}}$ (resp. the normalized Hamming metric $d_{n}^{\operatorname{Hamming}}$). Our main result is that if $Γ$ is infinite, hyperlinear (resp. sofic) and has Property $\operatorname{(T)}$, then it is not stable with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ (resp. $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$). This answers a question of Hadwin and Shulman regarding the stability of $\operatorname{SL}_{3}\left(\mathbb{Z}\right)$. We also deduce that the mapping class group $\operatorname{MCG}\left(g\right)$, $g\geq 3$, and $\operatorname{Aut}\left(\mathbb{F}_n\right)$, $n\geq 3$, are not stable with respect to $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$. Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on $\operatorname{U}\left(n\right)$ and the (unnormalized) $p$-Schatten metrics, since many groups with Property $\operatorname{(T)}$ are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ and $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$.

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