关于II_1因子的基本群和由群作用产生的等价关系
On the fundamental group of II_1 factors and equivalence relations arising from group actions
摘要：
给定一个可数群G，我们考虑正实数线上子群F的集合S_factor（G），S_eqrel（G），对它们存在一个自由遍历概率测度保持作用G，使得相应的II_1因子的基本群，轨道等价关系，等于F.证明了：若G是Z与非平凡群Γ的无穷多个拷贝的自由积，则S_factor（G）和S_eqrel（G）包含R_+本身、它的所有可数子群以及log在区间（0，1）内可具有任意豪斯多夫维数的不可数子群.证明了若G=Γ*Λ，其中Γ，Λ生成ICC群，且其中一个ICC群具有性质（T），则S_factor（G）=S_eqrel（G）={1}.我们还证明了存在II_1因子M使得M的基本群是R_+，但相应的II_ \infty因子M张量B（l^2）不允许R_+的连续迹标度作用.
英文摘要：
Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group Γ, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=Γ*Λ, with Γ, Λfinitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.
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