Related papers: Coarse structures on groups defined by conjugation…

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some…

A locally compact group $G$ is compact if and only if $L^1(G)$ is an ideal in $L^1(G)^{**}$, and the Fourier algebra $A(G)$ of $G$ is an ideal in $A(G)^{**}$ if and only if $G$ is discrete. On the other hand, $G$ is discrete if and only if…

For any locally compact group $G$ and any Banach algebra $A$, a characterization of the closed Lie ideals of the generalized group algebra $L^1(G,A)$ is obtained in terms of left and right actions by $G$ and $A$. In addition, when $A$ is…

If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of…

Let $\mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $\mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(H\cap…

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian…

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given…

Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $\Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the…

Let G be a connected reductive group defined over a non-archimedean local field of characteristic 0. We assume G is quasi-split, adjoint and absolutly simple. Let g be the Lie algebra of G. We consider the space of the invariant…

We define for discrete finitely presented groups a new property related to their asymptotic representations. Namely we say that a groups has the property AGA if every almost representation generates an asymptotic representation. We give…

We study lattices in a product $G = G_1 \times \dots \times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is…

In this note, we introduce a new concept of a {\it generalized algebraic rational identity} to investigate the structure of division rings. The main theorem asserts that if a non-central subnormal subgroup $N$ of the multiplicative group…

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; as well as an analogous relationship for spaces of left orders and…

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

For any finite group $G$, any transitive $G$-set $X$ and any field ${\Bbb F}$, we consider the vector space ${\Bbb F}^X$ of all functions from $X$ to ${\Bbb F}$, which is a $G$-space isomorphic to the permutation ${\Bbb F} G$-module ${\Bbb…

Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that…

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse…