Questions tagged [topological-groups]

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces. I am looking for classes of examples of connected topological groups/...
ThorbenK's user avatar
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Image of a complete topological group under open and surjective map is complete?

A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff. Question. Let $G$ be a complete topological group and let $H$ be a topological ...
Slup's user avatar
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Compact subgroups of a linear group over non-Archimedean local field

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
asv's user avatar
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4 votes
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Does every (Abelian) Polish group have a nontrivial locally compact subgroup?

The question is pretty much in the title, suppose that $G$ is an (Abelian) nontrivial Polish group, must $G$ have a nontrivial locally compact (in the induced topology, hence necessarily closed) ...
Alessandro Codenotti's user avatar
10 votes
1 answer
220 views

Classifying space of centralizer

$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let $$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$ be the homotopy ...
Thomas's user avatar
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Are closures of products of unimodular subgroups unimodular?

Let $G$ be a locally compact group, $N \subset G$ a unimodular normal subgroup, and $H \subset G$ a discrete (hence unimodular) subgroup. Does it follow that the closure $\overline{NH} \subset G$ is ...
Kim's user avatar
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Maximal protori in compact topological groups

I read the following proposition in the article in the link. Since $G$ itself is finite-dimensional, isn't the maximal protorus $T$ also finite-dimensional? In this case, $G$ becomes a Lie group. Isn'...
Mehmet Onat's user avatar
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Is the union of 1-dimensional pro-tori in a finite dimensional pro-torus dense?

Is the union of 1-dimensional compact connected abelian subgroups in a finite dimensional compact connected abelian group dense?
Mehmet Onat's user avatar
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Is $\mathbf{C}_p(X)$ self-dual?

Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
Luiz Felipe Garcia's user avatar
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Using the von Neumann crossed product to introduce a measure on the orbit space?

Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space). Question: is there a natural way of using the ...
Stepan Plyushkin's user avatar
8 votes
2 answers
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Is there a purely topological definition of $\text{Spin}(p,q)$?

I'm cross-posting this question from Math.SE, as it didn't get much attention there (even after a bounty). A common way to define the group $\text{Spin}(p,q)$ is via Clifford algebras. However, $\text{...
WillG's user avatar
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Topologies on diffeomorphisms groups

Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual ...
HUO's user avatar
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Can Rep(G) tell us whether G is discrete?

Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations. The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...
André Henriques's user avatar
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Continuous representations of topological groups on Banach spaces

Let $G$ be a topological group and let $\rho:G\rightarrow\text{GL}(V)$ be a linear representation of $G$ on a Banach space $V$. The representation is called strongly continuous if the map $g\mapsto\...
Botwinnik's user avatar
7 votes
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An abelian category with a full embedding from topological abelian groups

I know this is a very vague question, but I can't think of a better question to ask. Consider the category $\mathscr{C}$ of pro-sets created by diagrams containing only injections. The objects $A: \...
Charles Wang's user avatar
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Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question). Some simple ...
Pietro Majer's user avatar
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6 votes
2 answers
272 views

Uniqueness of left-invariant Borel probability measure on compact groups

On a compact topological group, consider two left-invariant probability measures $\mu$ and $\nu$ defined on the Borel sigma-algebra. Is it true that they coincide? It is classical that the Haar ...
Sebastien Gouezel's user avatar
2 votes
1 answer
218 views

Examples of non-discrete, cocompact subgroups

I am looking for non-trivial examples of the following: $G$ is a locally compact group $H\subset G$ a closed subgroup Both are unimodular and non-discrete The quotient space $G/H$ is compact, but $G$ ...
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Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]

Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$ i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$ Now let $X$ be the restriction of its ...
it's a hire car baby's user avatar
2 votes
0 answers
406 views

Generalized conjugacy classes in (topological) groups

Let $G$ be a topological group. We define an equivalence relation on $G$ as follows: For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate: $$x\mapsto ax,\qquad x\...
Ali Taghavi's user avatar
1 vote
0 answers
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Minimal F-semi-norms

There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...
erz's user avatar
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(Non)complete abelian groups in the “transfinite p-adic topology”

For an abelian group $A,$ a prime $p$ and an ordinal $\alpha,$ we recursively define $p^\alpha A$ as a subgroup of $A$ such that $p^0A=A,$ $$p^{\alpha+1}A=p(p^\alpha A) \hspace{5mm} \text{and} \...
Sergei Ivanov's user avatar
15 votes
2 answers
1k views

Does $\Bbb Z[X]$ determine $X$?

For a Hausdorff space $X$, consider the free abelian group $\mathbb{Z}[X]$ generated by $X$. Equip it with the finest topology which makes the map $X\to\mathbb{Z}[X]$, $x\mapsto [x]$ continuous and ...
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3 votes
0 answers
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Are U(H) and PU(H) locally uniform topological groups with the norm topology? Towards an instance of infinite-dimensional Hilbert's Fifth Problem

In looking at the work of Enflo generalising Hilbert's Fifth Problem from the Euclidean to the Banach case, there are the following conditions: the multiplication in the topological group is locally ...
David Roberts's user avatar
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109 views

The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple ...
Analyst's user avatar
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Idempotent conjecture and non-abelian solenoid

Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
Ali Taghavi's user avatar
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Idempotent conjecture and (weak) connectivity of (a reasonable) dual group

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space? The Motivation: The motivation comes from the idempotent conjecture of ...
Ali Taghavi's user avatar
2 votes
1 answer
201 views

Parametrization of topological algebraic objects

There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
erz's user avatar
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6 votes
0 answers
227 views

What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
Mark Schultz-Wu's user avatar
5 votes
0 answers
263 views

$T_1$ paratopological group having a dense commutative subgroup is commutative

I'm learning about topological groups from Arhangelskii and Tkachenko "Topological groups and related structures" and this is one of the exercises there. A paratopological group is a group ...
Jakobian's user avatar
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1 answer
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Non-continuous group homomorphism from p-adic field to C*

Let $F$ be a p-adic field. A character on $F^\times$ is defined as a continuous group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$. But is there any way to construct a non-continuous group ...
Windi's user avatar
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6 votes
1 answer
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Topologies that turn the real numbers into a compact Hausdorff topological group

If I'm not mistaken, the question I put on the title used to be on this site, but I'm not being able to find it at all. I'm therefore reposting it so that someone can either give me the old link or ...
Pedro Lourenço's user avatar
1 vote
1 answer
103 views

Continuity of central character [closed]

Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...
Windi's user avatar
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1 answer
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Is norm-continuous representation factored through a Lie quotient group?

I asked this 11 days ago at MSE, but there was no answer, I hope people here could help. Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is ...
Sergei Akbarov's user avatar
5 votes
0 answers
139 views

Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
Yanlong Hao's user avatar
1 vote
0 answers
43 views

Discreteness of $D^{-1}D$ given that $D$ is uniformly discrete

Let $G$ be a topological group with unit element $e$. We say that $D\subseteq G$ is discrete if for all $x\in D$ there is a unit-neighborhood $U\subseteq G$ such that $x^{-1}D\cap U=\{e\}$. We say ...
mathemagician99's user avatar
8 votes
2 answers
564 views

Homotopic but not equivariantly homotopic maps

Let $G$ be a topological (or simplicial) group, let $X$ and $Y$ be $G$-spaces, and let $f,f':X\to Y$ be $G$-maps which are homotopic as maps of spaces. In general, $f$ and $f'$ may (of course) fail to ...
Ken's user avatar
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5 votes
1 answer
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Extreme amenability of topological groups and invariant means

Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...
Muduri's user avatar
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10 votes
1 answer
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Are there extremally disconnected groups?

A Hausdorff space is called extremally disconnected or extreme, if for every open set $U$ the closure $\overline U$ is open, too. The question, whether there are extremally disconnected topological ...
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2 votes
1 answer
235 views

Does every locally compact group G contain a maximal open subgroup P which is a pro-Lie group?

EDIT 1: All topological groups in this question are assumed to be second countable. In particular, this forces every group to be metrizable and every Lie group to have at most countably many ...
Cosine's user avatar
  • 559
3 votes
1 answer
166 views

Open conjugacy classes in a second countable profinite group

Let $G$ be a second countable profinite group, $g\in G$ and $g^G:=\{hgh^{-1}~|~h\in G\}$ the conjugacy class of $g$ in $G$. Theorem 3.2 in Wesolek's Conjugacy class conditions in locally compact ...
stupid boy's user avatar
1 vote
0 answers
113 views

Density of irreducible matrix coefficients of a locally compact group

Let $G$ be a locally compact group and $I$ the set of matrix coefficient of irreducible unitary matrix coefficients of $G$. By Gelfand-Raikov's theorem and Stone-Weirestrass's theorem, for a compact $...
Pople's user avatar
  • 11
6 votes
1 answer
445 views

Groups with no (proper) closed subgroups?

$\mathbf{Z}$ with the profinite topology has the property that every subgroup is closed (Topological groups in which all subgroups are closed). What topological groups have the property that no (...
aaragon's user avatar
  • 73
2 votes
1 answer
212 views

Haar measures of compact subgroups

Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$: $$ \mu_K(K)=1. $$ Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
Sergei Akbarov's user avatar
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0 answers
73 views

Examples of amenable, Hausdorff, locally compact, second countable groups which are not discrete, not compact, and not abelian

I'm working on a problem that involves an amenable group acting on some set by bijections. Initially, I assumed the group was discrete and the set was countable, however I realized that the arguments ...
Jacob R's user avatar
  • 119
8 votes
2 answers
330 views

Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?

Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
Taras Banakh's user avatar
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1 vote
0 answers
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Neighborhoods of idempotents in topological inverse semigroups

In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
Bumblebee's user avatar
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1 vote
1 answer
177 views

Another question about unitary and anti-unitary matrices

This question is related to the previous post, "A question about unitary and anti-unitary matrices". Following the suggestion of Lspice, I am posting it as a separate question, as it might ...
jacaboul's user avatar
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6 votes
0 answers
68 views

Countable companions for Polish locally compact groups and their orbit equivalence relations

In "Countable sections for locally compact group actions" (Ergod. Th. & Dynam. Sys., 1992), Kechris proved that if $G$ is a Polish locally compact group acting in a Borel way on a ...
Iian Smythe's user avatar
  • 2,991
0 votes
0 answers
102 views

Classification of closures of additive subgroups of $\mathbb{R}^n$

If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either $\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or $\...
Nate Ackerman's user avatar

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