Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups

Continuous orbit equivalence rigidity

We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterise continuous orbit equivalence in terms of isomorphisms of C*-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Secondly, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this …

非奇异伯努利交叉积的分类结果

Classification results for nonsingular Bernoulli crossed products

本文针对自由群及其自由积群的非奇异伯努利作用，证明了 III 型因子的刚性与分类结果，首次在保概率测度情形之外给出无法用 Connes τ-不变量区分的大量互不同构 III$_1$ 型伯努利交叉积。

Strong Rigidity of II$_1$ Factors Arising from Malleable Actions of w-Rigid Groups, I

We consider cross-product II$_1$ factors $M = N\rtimes_σ G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $σ: G \to {\text{\rm Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as…

W*-superrigidity for Bernoulli actions of property (T) groups

We consider group measure space II$_1$ factors $M=L^{\infty}(X)\rtimesΓ$ arising from Bernoulli actions of ICC property (T) groups $Γ$ (more generally, of groups $Γ$ containing an infinite normal subgroup with relative property (T)) and prove a rigidity result for *--homomorphisms $θ:M\to M\bar{\otimes}M$. We deduce that the action $Γ\curvearrowright X$ is W$^*$--superrigid. This means that if $Λ\curvearrowright Y$ is {\bf any other} free, ergodic, measure preserving action such that the factors $M=L^{\infty}(X)\rtimesΓ$ and $L^{\infty}(Y)\rtimesΛ$ are isomorphic, then the actions $Γ\curvearro…

Tensor product decompositions and rigidity of full factors

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitr…

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equ…

大类型Artin群的刚性及分类结果

Rigidity and classification results for large-type Artin groups

本文计算了许多大类型Artin群交集图的自同构群，并以此为应用推导出包括外自同构群、可公度性分类及拟等距、测度等价、轨道等价、格嵌入和交叉积von Neumann代数刚性等一系列分类与刚性结果。

Bernoulli actions and infinite entropy

We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes the computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two genera…

New examples of W$^*$ and C$^*$-superrigid groups

A group $G$ is called $W^*$-superrigid (resp. $C^*$-superrigid) if it is completely recognizable from its von Neumann algebra $L(G)$ (resp. reduced $C^*$-algebra $C_r^*(G)$). Developing new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of $W^*$-superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in $C^*$-algebra theory including additional examples of $C^*$-superrigid groups and explicit computations o…

Explicit computations of all finite index bimodules for a family of II_1 factors

We study II_1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result: every finite index M-N-bimodule (in particular, every isomorphism between M and N) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index M-M-bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II_1 factor. We obtain in particular expli…