Related papers: Rigidity for graph product von Neumann algebras

We prove that the approximately inner automorphism group of a separable strongly stable von Neumann algebra is contractible in the u-topology. Thus the automorphism group of the hyperfinite type III_1 factor is contractible.

Developing new techniques at the interface of geometric group theory and von Neumann algebras, we identify the first examples of ICC groups $G$ whose von Neumann algebras are McDuff and exhibit a new rigidity phenomenon, termed McDuff…

In this paper, we show that for a graph $\Gamma$ from a class named H-rigid graphs, its subgraph ${\rm Int}(\Gamma)$, named the internal graph of $\Gamma$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors…

We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group \Gamma_1 * \Gamma_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced…

We study properties of automorphisms of graph products of groups. We show that graph product $\Gamma\mathcal{G}$ has non-trivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has…

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For $n\geq 3$ and a free ergodic probability measure preserving…

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We prove that the…

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In…

We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $\varphi_0$ and any discrete group $\Gamma$, the associated Bernoulli crossed product von…

von Neumann algebras have been playing an increasingly important role in the context of gauge theories and gravity. The crossed product presents a natural method for implementing constraints through the commutation theorem, rendering it a…

Quantum symmetry of graph $C^{*}$-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group $(\underbrace{C(S^{1})*C(S^{1})*\cdots…

We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete.…

The analytic von Neumann regular closure $R(\Gamma)$ of a complex group algebra $\C\Gamma$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(\Gamma)$ containing…

Let $(\Gamma,\mu)$ be a bipartite graph together with a weight on its vertices. Assume that $\mu$ is an eigenvector for the adjacency matrix of $\Gamma$. Let Aut$(\Gamma, \mu)$ be the automorphism group of the bipartite graph $\Gamma$ that…

We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If $\Gamma_1$, $\Gamma_2$ are two semi-arithmetic lattices in $\mathrm{PSL}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\colon\Gamma_1\to\Gamma_2$ is a group…

Let $\Gamma$ be a discrete group acting on a compact Hausdorff space $X$. Given $x\in X$, and $\mu\in\text{Prob}(X)$, we introduce the notion of contraction of $\mu$ towards $x$ with respect to unitary elements of a group von Neumann…

Given a connected large-type Artin group $A_\Gamma$, we introduce a deformation space $\mathcal{D}$. If $\Gamma$ is triangle-free, or has all labels at least 6, we show that this space is canonical, in that it depends only on the…

Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $\Gamma$ of $X$ is a free group and acts on the universal covering tree $\Delta$ and on its boundary $\partial \Delta$, endowed…