Related papers: Polynomial growth, comparison, and the small bound…

Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof…

Let a countable amenable group $G$ act on a \zd\ compact metric space $X$. For two clopen subsets $\mathsf A$ and $\mathsf B$ of $X$ we say that $\mathsf A$ is \emph{subequivalent} to $\mathsf B$ (we write $\mathsf A\preccurlyeq \mathsf…

Partial actions of discrete abelian groups can be used to construct both groupoid C*-algebras and partial crossed product algebras. In each case there is a natural notion of an analytic subalgebra. We show that for countable subgroups of…

We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…

We develop the notion of Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming…

We construct centrally large subalgebras in crossed products of $C (X, D)$ by automorphisms in which $D$ is simple, $X$ is compact metrizable, the automorphism induces a minimal homeomorphism of $X$, and a mild technical assumption holds.…

Given a locally compact group $G$, we study the smallest exact crossed-product functor $(A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G$ on the category of $G$-$C^*$-dynamical systems. As an outcome, we show that the smallest exact…

We define spectral freeness for actions of discrete groups on C*-algebras. We relate spectral freeness to other freeness conditions; an example result is that for an action of a finite group, spectral freeness is equivalent to strong…

Let X be an infinite, compact, metrizable space of finite covering dimension and h a minimal homeomorphism of X. We prove that the crossed product of C(X) by h absorbs the Jiang-Su algebra tensorially and has finite nuclear dimension. As a…

Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action…

We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither…

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct…

Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its $C^{*}$-algebra is isomorphic to a crossed product.

In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius $n$ centered at the identity. We prove that in the case of virtually polycyclic groups, this…

It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such…

We develop a dynamical version of some of the theory surrounding the Toms-Winter conjecture for simple separable nuclear C*-algebras and study its connections to the C*-algebra side via the crossed product. We introduce an analogue of…

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if…

We decompose the crossed product functor for actions of crossed modules of locally compact groups on C*-algebras into more elementary constructions: taking crossed products by group actions and fibres in C*-algebras over topological spaces.…

We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to…

In this paper we show that the Cuntz--Pimsner algebras associated to minimal homeomorphisms twisted by line bundles, along with their orbit-breaking subalgebras, are $\mathcal{Z}$-stable whenever the underlying dynamical system has the…