Related papers: Shift-minimal groups, fixed price 1, and the uniqu…

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining…

We show that any countable subgroup of the multiplicative group $\mathbb{R}_+^{\times}$ of positive real numbers can be realized as the fundamental group $\mathcal{F}(A)$ of a separable simple unital $C^*$-algebra $A$ with unique trace.…

We prove that, for a free action $\alpha \colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $\Gamma$ of the Cartan subalgebra…

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A \rtimes \Gamma covering certain cases where \Gamma is an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we…

We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli.…

Let $\mathcal{G}$ be a locally compact Hausdorff \'{e}tale groupoid. We call a tracial state $\tau$ on a general groupoid $C^*$-algebra $C_\nu^*(\mathcal{G})$ canonical if $\tau=\tau|_{C_0(\mathcal{G}^{(0)})} \circ E$, where…

For \Gamma a countable amenable group consider those actions of \Gamma as measure-preserving transformations of a standard probability space, written as {T_\gamma}_{\gamma \in \Gamma} acting on (X,{\cal F}, \mu). We say…

We describe two kinds of regular invariant measures on the boundary path space of a second countable topological graph, which allows us to describe all extremal tracial weights on the graph C$^{*}$-algebra which are not gauge-invariant.…

A classical construction associates to a transient random walk on a discrete group $\Gamma$ a compact $\Gamma$-space $\partial_M \Gamma$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M \Gamma)…

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show…

Let $\Gamma$ be a countable group and $\mathrm{Sub}(\Gamma)$ its Chabauty space, namely the compact $\Gamma$-space consisting of all subgroups of $\Gamma$. We call a subgroup $\Delta \in \mathrm{Sub}(\Gamma)$ a boomerang subgroup if for…

For a finite (not necessarily Abelian) group $(\Gamma,\cdot)$, let $n(\Gamma) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from…

A discrete countable group \Gamma is said to be ME rigid if any discrete countable group that is measure equivalent to \Gamma is virtually isomorphic to \Gamma. In this paper, we construct ME rigid groups by amalgamating two groups…

A standard tool for classifying the complexity of equivalence relations on $\omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce…

We prove simplicity of all intermediate $C^*$-algebras $C^*_{r}(\Gamma)\subseteq \mathcal{B} \subseteq \Gamma\ltimes_r C(X)$ in the case of minimal actions of $C^*$-simple groups $\Gamma$ on compact spaces $X$. For this, we use the notion…

I. M. Chiswell has asked whether every group that admits a free isometric action (without inversions) on a $\Lambda$-tree is orderable. We give an example of a multiple HNN extension $\Gamma$ which acts freely on a $\mathbb{Z}^2$-tree but…

We approach the study of sub-von Neumann algebras of the group von Neumann algebra $L\Gamma$ for countable groups $\Gamma$ from a dynamical perspective. It is shown that $L(\Gamma)$ admits a maximal invariant amenable subalgebra. The notion…

In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.

As an analogue of the topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M, \tau)$ and apply it to generalize the main results of [AHO23], showing that for a…

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is…