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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Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected?

I was reading this question The connected component of the idele class group but I am very confused about the structure of the solenoids $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\...
Yuan Yang's user avatar
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1 answer
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Density of matrix coefficients of unitary representations of a locally compact group

Let $G$ be a locally compact group, $C_0(G)$ the $C^*$-algebra of continuous functions on $G$ that vanish at infinity, $C_b(G)$ the $C^*$-algebra of bounded continuous functions on $G$. We know that $...
Rick Sternbach's user avatar
5 votes
0 answers
163 views

Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$

Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$? Apparently, there is an almost complete classification in ...
Arshak Aivazian's user avatar
2 votes
1 answer
221 views

Show that $V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$ defined by $V_G(f)(x,y) = f(xy,y)$ is well-defined

Let $G$ be a locally compact Hausdorff group and let $\mu$ be a right Haar measure on $G$. Then $\mu\times \mu$ (the Radon product of measures) is a right Haar measure on $G \times G$ and we can ...
Andromeda's user avatar
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14 votes
3 answers
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Examples of locally compact groups that do not admit enough finite dimensional representations

I apologize in advance if this is well-known, but I can't seem to find the answer in the literature. Let me be precise about my question. I am looking for concrete examples of locally compact ...
Rick Sternbach's user avatar
10 votes
0 answers
254 views

What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?

What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$ I know that neither ...
Alessandro Codenotti's user avatar
7 votes
2 answers
318 views

If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?

We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is ...
wonderich's user avatar
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4 votes
1 answer
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The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...
Muduri's user avatar
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2 answers
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Group actions on affine varieties with closed orbits

The following is motivated by a (now-deleted) MSE-question by @aglearner. Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the ...
Moishe Kohan's user avatar
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8 votes
1 answer
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Topological group locally homeomorphic to the Hilbert cube

Does there exist a topological group which is locally homeomorphic to the Hilbert cube $[0,1]^{\mathbb N}$? Let me note that Hilbert cube has the fixed point property and thus it is not homeomorphic ...
Benjamin Vejnar's user avatar
2 votes
1 answer
365 views

Topology of multiplication groups of local fields

In Neukirch’s book “Algebraic Number Theory”, Proposition II.5.7, the following is insisted: for a mixed characteristic local field $K$ with a residue field $\mathbb{F}_q$, $q = p^f$, then one has an ...
Alice's user avatar
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0 votes
0 answers
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Is second countability an extension property for non-Hausdorff spaces?

Let $G$ be an abelian topological group and let $H$ be a non-Hausdorff closed subgroup (so that $G/H$ is Hausdorff). If $H$ and $G/H$ are second countable, is $G$ second countable?
Cristian D. Gonzalez-Aviles's user avatar
2 votes
1 answer
160 views

Gelfand-Naimark and Peter-Weyl for the unitary group

Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}_l$. Which means that they both have ...
Jake Wetlock's user avatar
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20 votes
0 answers
301 views

Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
Calculix's user avatar
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1 vote
0 answers
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Topology of direct product of $ F_{p}$

Let's assume that $F$ is a number field, $R^{*}$ =$\prod_{P\in S}$$F_{P}$, $R$ is the Adele ring of $F$ and $$ R' = \prod_{P\in S_{0}}\mathfrak{o}_{\mathfrak{p}}\times\prod_{P\in S_{\infty}}F_{P}.$$ ...
Fuutorider's user avatar
3 votes
0 answers
91 views

About the nilpotency of a subgroup

Let $G$ be a compact group. Let $\mathcal N$ be a family of closed normal subgroups of nilpotency class at most $k$. Assume that $\mathcal N$ is closed under finite intersections and $\bigcap_{N\in\...
Meisam Soleimani Malekan's user avatar
3 votes
1 answer
366 views

Principal bundles from a fibration of homogeneous spaces

Let $G$ be a compact (Lie) group, and $H \subseteq H'$ two compact (Lie) subgroups. It is clear that we have an obvious surjective map of homogeneous spaces $$ G/H \twoheadrightarrow G/H'. $$ Will it ...
Spyros Olympopolous's user avatar
5 votes
1 answer
152 views

Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?

Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that $$\...
Taras Banakh's user avatar
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3 votes
1 answer
158 views

Bounds for a covering number of the circle group $\mathbb T$ by some its small subgroups

$\newcommand{\w}{\omega}\newcommand{\A}{\mathcal A}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}...
Alex Ravsky's user avatar
  • 4,102
0 votes
1 answer
173 views

Fourier transform on lattice strip

I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin ...
spaceman's user avatar
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5 votes
1 answer
281 views

Are nearby crossed homomorphisms from compact Lie groups crossed-conjugate?

Charles Rezk had highlighted in MO:q/123624 that "Nearby homomorphisms from compact Lie groups are conjugate", and in consequence -- further highlighted in Remark 2.2.1 of his Global ...
Urs Schreiber's user avatar
6 votes
1 answer
264 views

A property of rapid sequences of natural numbers

$\newcommand{\IR}{\mathbb R}$ $\newcommand{\IT}{\mathbb T}$ $\newcommand{\w}{\omega}$ $\newcommand{\e}{\varepsilon}$ Taras Banakh and me proceed a long quest answering a question of ougao at ...
Alex Ravsky's user avatar
  • 4,102
6 votes
1 answer
304 views

Topology on cohomology of a sheaf of topological groups

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question: Is there a natural way to introduce topology on $H^i(X,...
 V. Rogov's user avatar
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6 votes
2 answers
372 views

About Lie group $G$ has this escape property?

Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$. ...
free's user avatar
  • 61
15 votes
1 answer
460 views

For what LCH groups is the Haar measure $\mu(U x U)$ bounded?

Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function $$ \Phi: \quad G \to (0,\infty), \quad x \...
PhoemueX's user avatar
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1 vote
1 answer
369 views

Characters of p-adic units

Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...
Osheaga's user avatar
  • 59
5 votes
0 answers
218 views

When is the profinite completion of a Noetherian group ring also Noetherian?

Let $G$ be a group, and let $\mathbb{Z}[G]$ denote its group ring. Its profinite completion is the inverse limit over all ideals of finite index. By Benjamin Steinberg's answer here, this profinite ...
stupid_question_bot's user avatar
6 votes
2 answers
326 views

Intersection of all open subgroups vs. the intersection of all open normal subgroups

I am interested to know examples of topological groups $G$ for which the intersection $\bigcap\{H\leq G\mid H\text{ open}\}$ of all open subgroups of $G$ is the trivial subgroup but for which the ...
Jeremy Brazas's user avatar
7 votes
1 answer
207 views

Why are free Boolean topological groups Hausdorff?

Assume $X$ is a Tychonoff space. Then $A(X)$ is the free topological abelian group over $X$. I know that $A(X)$ is Hausdorff and the canonical embedding from $X$ to $A(X)$ is a topological embedding. ...
Sevim's user avatar
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6 votes
2 answers
778 views

Is every continuous action of a compact topological group closed?

I am reading Bredon's Introduction to compact transformation groups, and came across the following result and proof on page 34: Even though he writes "Recall our standing assumption that $X$ is ...
Ben's user avatar
  • 1,010
5 votes
1 answer
242 views

How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?

For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
Alessandro Codenotti's user avatar
1 vote
0 answers
135 views

Terminology for an kind-of principal fibration

My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets. Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
Jeff Strom's user avatar
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4 votes
1 answer
274 views

Finite covolume of uniform lattice in quotient group

Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact. Suppose ...
user avatar
3 votes
0 answers
123 views

Restrictions on pointed lifts of isometries

Let $M$ be a (closed) Riemannian manifold and let $f$ be an isometry of $M$ fixing a point $\ast \in M$ that acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ of ...
Jens Reinhold's user avatar
8 votes
1 answer
467 views

Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig

"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
მამუკა ჯიბლაძე's user avatar
1 vote
0 answers
30 views

Morphism of R-algebras between R-adic algebras

Let $R$ be a $f$-adic ring. Let $A$ and $B$ be $R$-adic algebras. I would like to show that any morphism of $R$-algebras between $A$ and $B$ is actually adic.
Pierre21's user avatar
  • 385
1 vote
1 answer
548 views

Haar measure coming from Pontryagin duality v/s Fourier inversion

Not research but advertising this question from mse in case someone wants to answer. I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
Chertopkhanov on Malek Adel's user avatar
2 votes
0 answers
78 views

Definition of a continuous Gabor frame

I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\...
Jake Wetlock's user avatar
  • 1,114
0 votes
0 answers
134 views

Left-side cosets of an open subgroup

Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}\in G$. We assume $L\cap g_{1}K\neq \emptyset$ and $L\cap g_{2}...
M masa's user avatar
  • 479
12 votes
0 answers
331 views

Metric completion of an algebraically closed field is algebraically closed?

Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed? We can ...
Gerald Edgar's user avatar
  • 40.2k
1 vote
1 answer
146 views

Do Locally Contractible, Path-Connected Groups have Accessible Bases?

Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\...
John Samples's user avatar
5 votes
1 answer
842 views

Countable sum $\bigoplus_{n=0}^\infty\mathbb Z_p$ as a topological group

$\DeclareMathOperator\colim{colim}$This is inspired by Clausen's answer. Question: Recall that $\mathbb Z_p$ is endowed with the $p$-adic topology. Consider the countable sum $M:=\bigoplus_{n=0}^\...
Z. M's user avatar
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5 votes
0 answers
187 views

Haar mesure on $\mathrm{GL}_{d}(F)$

$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as $$ A= \left( \begin{array}{ccc} a_{11}+t^{\...
M masa's user avatar
  • 479
1 vote
1 answer
249 views

CH and the density topology on $\mathbb{R}$

In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
Gabriel Medina's user avatar
1 vote
0 answers
277 views

On measurability of certain group actions on spaces of bounded measurable functions

Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded ...
S.Z.'s user avatar
  • 463
3 votes
0 answers
148 views

Is there a Hausdorff space that is also a group such that the group operation is continuous but the inversion map is not continuous?

The question is from the definition of to topological group. I can find an example such that the inversion map is continuous but the group operation is not continuous, but I cannot find an example ...
Zyis's user avatar
  • 31
8 votes
1 answer
248 views

Pointed versus unpointed maps into a topological monoid

I've just stumbled on something that seems either too good to be true, or else too good for me not to have heard of it before. It has to do with the basepoint forgetting map $$ u: [A, M] \to \langle A,...
Jeff Strom's user avatar
  • 12.5k
8 votes
1 answer
456 views

About locally compact groups without compact subgroups

Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
Wlod AA's user avatar
  • 4,686
9 votes
2 answers
804 views

Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean?

Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I ...
Adam's user avatar
  • 2,370
6 votes
1 answer
165 views

With a linear representation, how does the continuity of $G \to \mathrm{GL}(V)$ relate to that of $G \times V \to V$?

I'm currently reading Traces of Hecke Operators by Knightly and Li, while simultaneously revisiting the adelic/representation-theoretic point of view on automorphic forms. In Knightly and Li, they ...
davidlowryduda's user avatar

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