arXiv 1909.002822019-09math.GRmath.OA

产品组和特性$(τ)$的稳定性

Stability for product groups and property $(τ)$

Adrian Ioana

摘要：

研究可数群的置换稳定性（或P-稳定性）。我们的主要结果提供了一个广泛的一类不适合的产品组，这不是P-稳定的。这个类包括积群$Σ\timesΛ$，只要$Σ$允许非阿贝尔自由商，并且$Λ$允许无限循环商。特别地，我们得到了群$\mathbb F_m\times\mathbb Z^d$和$\mathbb F_m\times\mathbb F_n$对任意整数$m,n\geq 2$和$d\geq 1$都不是P-稳定的.这意味着P-稳定性在直积构造下不是封闭的，这回答了Becker，Lubotzky和Thom的一个问题。我们的主要结果的证明依赖于从$Σ\timesΛ$到有限对称群的渐近同态的构造，从具有和不具有性质$(τ)$的$Σ$和$Λ$中的有限指数子群的序列开始。我们的方法是足够强大的，以表明所涵盖的群体甚至没有灵活的P-稳定，从而给出了第一个这样的不服从剩余有限的例子。

英文摘要：

We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group $Σ\timesΛ$, whenever $Σ$ admits a non-abelian free quotient and $Λ$ admits an infinite cyclic quotient. In particular, we obtain that the groups $\mathbb F_m\times\mathbb Z^d$ and $\mathbb F_m\times\mathbb F_n$ are not P-stable, for any integers $m,n\geq 2$ and $d\geq 1$. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. The proof of our main result relies on a construction of asymptotic homomorphisms from $Σ\timesΛ$ to finite symmetric groups starting from sequences of finite index subgroups in $Σ$ and $Λ$ with and without property $(τ)$. Our method is sufficiently robust to show that the groups covered are not even flexibly P-stable, thus giving the first such non-amenable residually finite examples.

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