Related papers: From isolated subgroups to generic permutation rep…

Let $G$ be a finitely generated group, $\mathrm{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chaubuty topology and $X!$ the (Polish) group of all permutations of a countable set $X$. We show that the following…

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $\Gamma$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{Rep}(\Gamma,G)$ of all representations of $\Gamma$ in…

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group $S_\infty$, the automorphism group of the countable dense linear order, the…

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that…

For a connected quasi-split reductive algebraic group $G$ over a field $k$, which is either a finite field or a non-archimedean local field, $\theta$ an involutive automorphism of $G$ over $k$, let $K =G^\theta$. Let $K^1=[K^0,K^0]$, the…

Let $G$ be a discrete group with property (T). It is a standard fact that, in a unitary representation of $G$ on a Hilbert space $\mathcal{H}$, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by…

Let $\psi : G\to GL(V)$ and $\varphi :G \to GL (W)$ be representations of finite group $G$. A linear map $T: V\to W$ is called a morphism from $\psi$ to $\varphi$ if it satisfys $T\psi_g= \varphi_g T$ for each $g\in G$ and let…

A group G is representable in a Banach space X if G is isomorphic to the group of isometries on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases,…

If $G$ is a finite group or a torus, it is known that there is an isomorphism between the Grothendieck group of homotopy representations and that of generalized homotopy representations for $G$. We prove that there is such an isomorphism…

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We…

The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations…

A base of a permutation group (X,G) is a subset B of X such that its pointwise stabilizer is the trivial group. A list (x1,x2, ... ,xk) of elements of X is irredundant if each element is not in the pointwise stabilizer of its predecessors.…

Let $G$ be a group and let $\rho\colon G\to\operatorname{Sym}(V)$ be a permutation representation of $G$ on a set $V$. We prove that there is a faithful $G$-coalgebra $C$ such that $G$ arises as the image of the restriction of…

Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a…

Let G be a closed subgroup of the group of all permutations of a countably infinite set. Let X be a Polish G-space with a countable basis A of clopen sets. Each x from X defines a characteristic function f on A by f(U)=1 iff x belongs to U…

We consider the following class of unitary representations $\pi $ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau $, and that the Hilbert space…

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ over an…

Let $(\mathcal{G},\nu)$ be a $t$-discrete ergodic groupoid. Consider a finite Von Neumann algebra $\mathcal{M}$ with separable predual. We prove that every uniformly bounded measurable representation $\rho:\mathcal{G} \rightarrow…

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a…