Related papers: Precompact groups and property (T)

Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ on $G$ under which $F$ converges to $0$ has been described by I.Protasov and E.Zelenyuk. In particular, their description…

We introduce and study the class of groups graded by root systems. We prove that if {\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\Phi}, then under certain natural conditions on the grading,…

Every discrete group with Kazhdan's Property (T) is a quotient of a torsion-free, word hyperbolic group with Property (T).

A topological group $G$ is called an $M_\omega$-group if it admits a countable cover $\K$ by closed metrizable subspaces of $G$ such that a subset $U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every $K\in\K$. It is…

We prove that a discrete quantum group $\mathbb{G}$ has the approximation property if and only if a Fej\'{e}r-type representation holds for its $C^*$-algebraic or von Neumann algebraic crossed products. As applications, we extend several…

The aim of this paper is to establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids in the sense of P. Deligne and A. Brugui\`eres. In one direction we have the…

We introduce a non commutative analog of the Bohr compactification. Starting from a general quantum group G we define a compact quantum group bG which has a universal property such as the universal property of the classical Bohr…

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{\omega_1 \omega}$-axiomatization of amenability. We also show that in the case of…

In this paper we describe a classifying theory for families of simplicial topological groups. If $B$ is a topological space and $G$ is a simplicial topological group, then we can consider the non-abelian cohomology $H(B,G)$ of $B$ with…

In this paper, we study the uniformities on the double coset spaces in topological groups. As an implication, the quotient spaces of topological groups with a $q$-point are studied. It mainly shows that: (1) Suppose that $G$ is a…

By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In this paper, we present the $z$-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group $G$, let…

In this paper we investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a…

It is shown that each pseudonorm defined on a subgroup $H$ of an abelian group $G$ can be extended to a pseudonorm on $G$ such that the densities of the obtained pseudometrizable topological groups coincide. We derive from this that any…

Let $G$ be a reductive group over a nonarchimedean local field $F$. In the quest for a classification of irreducible smooth representations of $G$, it is critical to understand the case of supercuspidal representations -- those whose matrix…

Diffeomorphism groups $G$ of manifolds $M$ on locally $\bf F$-convex spaces over non-Archimedean fields $\bf F$ are investigated. It is shown that their structure has many differences with the diffeomorphism groups of real and complex…

We exhibit a topological group $G$ with property (T) acting non-elementarily and continuously on the circle. This group is an uncountable totally disconnected closed subgroup of $\operatorname{Homeo}^+(\mathbf{S}^1)$. It has a large unitary…

It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian…

Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this…

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more…

Let $G$ and $T$ be topological groups, $\alpha : T \to \Aut(G)$ a homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G \rtimes_\alpha T$ the corresponding semidirect product group. In this paper we address several…