关于由w-刚性群的2-上循环引起的II $_1$因子
On II$_1$ factors arising from 2-cocycles of w-rigid groups
摘要：
我们考虑$\text{\rm II}_1$因子$L_μ(G)$产生于群$G$上的2-cocyles $μ\in \text{\rm H}^2(G,\Bbb T)$，群$G$包含具有相对性质$\text{\rm(T)}$的无限正规子群$H \subset G$（即$G$ { \it w-刚性}）。证明了给定任意可分的$\text{\rm II}_1$因子$M$，具有$L_μ(G)$可嵌入$M$的性质的2-上圈集$μ_{|H}\in \text{\rm H}^2(H,\Bbb T)$至多可数.我们使用这一结果，相对属性（T）的$\Bbb Z^2 \subset \Bbb Z^2 \rtimes Γ$的$Γ\subset SL(2,\Bbb Z)$不顺从的事实，每一个上循环$μ_α\in {\text{\rm H}}^2(\Bbb Z^2,\Bbb T)\simeq \Bbb T$延伸到一个上循环的$\Bbb Z^2 \rtimes SL(2,\Bbb Z)$，表明一个参数家庭的II $_1$的因素$M_α(Γ)=L_{μ_α}(\Bbb Z^2 \rtimes Γ)$，$α\in \Bbb T$，是相互不同构，模可数集，并不能全部嵌入到同一个可分离的II $_1$的因素。其他的例子和应用进行了讨论。
英文摘要：
We consider $\text{\rm II}_1$ factors $L_μ(G)$ arising from 2-cocyles $μ\in \text{\rm H}^2(G,\Bbb T)$ on groups $G$ containing infinite normal subgroups $H \subset G$ with the relative property $\text{\rm(T)}$ (i.e. $G$ {\it w-rigid}). We prove that given any separable $\text{\rm II}_1$ factor $M$, the set of 2-cocycles $μ_{|H}\in \text{\rm H}^2(H,\Bbb T)$ with the property that $L_μ(G)$ is embeddable into $M$ is at most countable. We use this result, the relative property (T) of $\Bbb Z^2 \subset \Bbb Z^2 \rtimes Γ$ for $Γ\subset SL(2,\Bbb Z)$ non-amenable and the fact that every cocycle $μ_α\in {\text{\rm H}}^2(\Bbb Z^2,\Bbb T)\simeq \Bbb T$ extends to a cocycle on $\Bbb Z^2 \rtimes SL(2,\Bbb Z)$, to show that the one parameter family of II$_1$ factors $M_α(Γ)=L_{μ_α}(\Bbb Z^2 \rtimes Γ)$, $α\in \Bbb T$, are mutually non-isomorphic, modulo countable sets, and cannot all be embedded into the same separable II$_1$ factor. Other examples and applications are discussed.
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