# 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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### 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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### 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 ...

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### 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 ...

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### Problem concerning Haar measures on locally compact Hausdorff groups [migrated]

Before I state what my problem is I first want to give some context.
A Haar measure is a measure on the Borel subsets of a locally compact Hausdorff group $X$. The Haar measure is inner regular on ...

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### 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 $...

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### 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 (...

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### 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 ...

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### 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 ...

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### 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 $...

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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 ...

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### 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 ...

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### 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 ...

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### 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
$\...

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### What's the relation between pseudo-compact and admissible rings?

We recall two definitions. Let $A$ be a linearly topologized ring which is complete and Hausdorff.
We say that $A$ is pseudo-compact if, for every open ideal $I\subset A$, the ring $A/I$ is artinian. ...

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### Defining the classifying space of a group acting on a set

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts
on $n+1$-...

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### G separable group, $\aleph_0 \leq \tau$. What is $l(X)$ and $\omega l(X) (\leq \tau)$? where $X \subseteq G$. And what is $\chi (G)$ (cardinal)?

Happy Chinese new year!
I was reading (and translating) a Russian article "On the topological groups close to being Lindelöf".
Where it is assumed G is a separable group and $\tau \geq \...

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### On inverse limits of $\pi$-adically complete algebras

Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...

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### Theory of group representation for compact groups

I write here because some experts could help on that. It is very well known (at least for me) many reference books on linear representations of finite groups (for instance, the very classical and ...

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### Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?

Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...

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### Kazhdan's property $T$ implies not virtually indicable

I've read in many places that every compactly generated group $G$ satisfying Kazdhan's property (T) is not virtually indicable (there is no subgroup $H\leq G$ of finite index which surjects onto $\...

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### Is the left-regular representation of a locally compact group a homeomorphism onto its image?

Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda_g f(h) = f(g^{-1}h)$, for a locally compact group.
It is well-known that this is a unitary faithful and strongly-...

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### "Homotopy homomorphisms" of homeomorphisms of Euclidean space

For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...

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### Is every compact quasisimple group a Lie group?

Let $ G $ be a compact topological group which is quasisimple in the sense that
$$
[G,G]=G
$$
and
$$
G/Z(G)
$$
is simple as an abstract group. Must $ G $ be a Lie group?
This is a follow-up question ...

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### Openness of product of two open subgroups

Let $G$ be a profinite topological group with two closed subgroup $G_1$ and $G_2$. Suppose $G_1$ is normal in $G$ and $G=G_1G_2$. Let $H_i$ be an open subgroup in $G_i$ for $i=1,2$.
Question: Is $ ...

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### Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$

Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...

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### Norm under Gelfand map vs norm under left regular representation on $\ell^p$

Let $G$ be a discrete commutative group. Let $p \in [1,\infty)$ and consider the left regular representation $\lambda : \ell^1(G) \to \mathcal{B}(\ell^p(G))$; that is $\lambda(x)y := x*y$,
where
$$
(x*...

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### Continuity of "inversion operator" between function spaces

Question: When is the operation of inversion continuous as a map between spaces of invertible functions?
Let $\mathcal{F}$ be a function space such that $f\in\mathcal{F}\implies$ $f$ is invertible and ...

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### All-set-homogeneous spaces

This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property?
A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...

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### Constructing Complicated Borel Subgroups of Polish Groups

Farah and Solecki showed the following in Borel subgroups of Polish groups:
Theorem: Every Polish group $G$ admits Borel subgroups of arbitrarily high Borel rank.
However, the construction is far from ...

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### Alternative uniformities on topological groups

Are there any interesting alternative uniformities defined on topological groups besides the usual four (left, right, and their meet/join)? I am curious because in the (sort of) dual setting of coarse ...

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### Stone–Čech compactification as a semigroup

Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...

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### Steinhaus number of a group

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$.
Let $\mathcal A_X$ be the family of ...

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### Morphism in commutative square strict?

Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...

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### Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup

A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\...

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### Structure of a profinite group as a condensed set with an action of an open subgroup

Let $G$ be a profinite group and $H$ be an open subgroup. As a continuous $H$-topological space, we have $G=\coprod_{G/H} H$. Does this also hold as condensed sets, i.e. do we have an identification ...

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### Locally compact + two-point homogeneous => Riemannian

A metric space $M$ is called two-point homogeneous if for any pair of points $(p,q)$ in $M$ any distance preserving map $f\colon\{p,q\}\to M$ can be extended to an isometry $\bar f\colon M\to M$.
The ...

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### What are all of the topological (commutative) monoid structures on a closed interval?

Consider a closed real interval $[a,b]$ as a toplogical space. Up to homeomeorphism it doesn't matter, but I like to take $[a,b] = [0,\infty]$.
Question 1: What are all of the topological commutative ...

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### Willis theory for discrete groups?

Two important tools in the study of totally disconnected locally compact groups, introduced by George Willis, are the scale function and tidy subgroups. In principle, these notions are well-defined ...

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### What is the status of topological problems in group rings?

I have extensively studied group rings and structure of their units and also zero divisors and normalizer problem in integral group rings.. I was pondering upon questions like what if we use ...

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### Topological semi-direct products of groups

In Kaniuth, Taylor, Induced representations of locally compact groups on pages 9-10 it's claimed that if $G$ is a locally compact group with closed subgroups $N,H$, with $N$ normal in $G$, with $N\cap ...

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### Tensoring with an induced representation: proof question

Let $G$ be a locally compact Hausdorff group and $H$ a closed subgroup of $G$. If $\sigma: H \to B(\mathcal{K}_\sigma)$ is a unitary representation of $G$, we can associate an "induced ...

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### $\sigma$-compactness of some locally compact Hausdorff topological groups

Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology?
More generally, I'm looking for a ...

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### Urysohn's lemma for Bochner functions?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:
If $U$ is an open ...

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### Hausdorff distance in compact Lie groups

Let $G$ be a compact Lie group with a compatible biinvariant metric $d$. The hyperspace $K(G)$ of nonempty compact subsets of $G$ is a compact metric space with the Hausdorff metric, and it is easy to ...

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### Initial topology for a topological ring

Given a topological ring $R$ and an arbitrary (thus not necessarily surjective) epimorphism $q: R \to S$ of underlying rings is there a finest topology on $S$ such that 1) $S$ is a topological ring ...

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### A question about Marino–Prodi perturbation

In this paper N. Ghoussoub,
the author claims the following version of Marino–Prodi perturbation, that is :
Let $H$ a Hilbert space.
Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...

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### Compact Lie groups as quotients of torsion-free compact metrizable groups

The question:
(1) Is every compact Lie group $G$ isomorphic (as a topological group) to some quotient $H/N$ where $H$ is a torsion-free compact metrizable group?
Or equivalently:
(2) Is every compact ...

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### Solid tensor product of pro-discrete space with Laurent series

Consider the category $\operatorname{Solid}_{\mathbf{Z}}$ of solid abelian groups in the sense of Clausen-Scholze. This category is a full subcategory of condensed abelian groups, $\operatorname{Cond}...

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### Pontryagin-reflexivity of spaces of continuous functions

It is well-known that a Banach space $V$ is always Pontryagin-reflexive, i.e. the natural map $V\to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(V, \mathbb{R}), \mathbb{R})$, where $\text{Hom}_\mathbb{...

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### Skeletal topological groupoid

Let $G$ be a topological groupoid with the property that any two isomorphic objects are topologically indistinguishable in $\mathrm{Ob}(G)$. Does that imply that $G$ is equivalent to a skeletal ...