Related papers: Glasner's problem for Polish groups with metrizabl…
Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all…
A Poisson system is a Poisson point process and a group action, together forming a measure-preserving dynamical system. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We consider…
We answer an open question of Grigorchuk and Zuk about amenability using random walks. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct…
In this paper we show that for every congruent monotileable amenable group $G$ and for every metrizable Choquet simplex $K$, there exists a minimal $G$-subshift, which is free on a full measure set, whose set of invariant probability…
For each ordinal $\alpha<\omega_1$, we introduce the class of $\alpha$-balanced Polish groups. These classes form a hierarchy that completely stratifies the space between the class of Polish groups admitting a two-side-invariant metric…
Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every nontrivial element belongs to a generating pair. Groups with…
We begin the systematic study of decision problems for finitely generated groups given by a solution to their word problem. We relate this to the study of computable analysis on the space of marked groups. We point out that several distinct…
We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product…
We consider a finitely generated group acting minimally on a compact space by homeomorphsims, and assume that the Schreier graph of at least one orbit is quasi-isometric to a line. We show that the topological full group of such an action…
A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a…
We present a survey of results related to the Milnor's problem on group growth. We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but the main part of the article is devoted to the intermediate…
We show that an \'etale groupoid which is strongly amenable at infinity is amenable whenever its full and reduced $C^*$-algebras coincide.
If G is a Polish group, then there is a Polish G-space X which is universal among Polish G-spaces with respect to continuous G-embeddings.
We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],\lambda)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely…
Returning to a classical question in Harmonic Analysis we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers Z which is not in the norm-closure of the algebra B(Z)…
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of…
Let $G$ be a Polish (i.e., complete separable metric topological) group. Define $G$ to be an algebraically determined Polish group if for any Polish group $L$ and algebraic isomorphism $\varphi: L \mapsto G$, we have that $\varphi$ is a…
For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the…
Although NP-Complete problems are the most difficult decisional problems, it is possible to discover in them polynomial (or easy) observables. We study the Graph Partitioning Problem showing that it is possible to recognize in it two…
It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere $\s_{\cal H}$ has a…