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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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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Does there exist a Radon measure of full support on a non locally compact group that is translation invariant under some element?

It is known that a topological group is locally compact if and only if it has a Haar measure. However, I am curious about being translation invariant under at least one element. Does there exist a ...
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4 votes
1 answer
96 views

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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3 votes
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101 views

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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1 answer
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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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2 votes
1 answer
59 views

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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5 votes
1 answer
432 views

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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2 votes
2 answers
150 views

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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3 votes
0 answers
84 views

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 ...
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4 votes
2 answers
259 views

Topological groupoids and equivariant sheaves

Some statements that are true for ordinary groupoids fail for topological groupoids (by which I mean groupoids internal to the category of topological spaces): for instance, every ordinary groupoid is ...
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10 votes
2 answers
536 views

Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?

Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
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5 votes
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Are isomorphic subgroups of the symmetric group conjugate in some larger group?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}$Let $S_\infty$ be the group of all permutations of a countable infinite set (enriched with the product Polish topology). Does there exist a ...
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1 vote
1 answer
181 views

Approximations by compact sub-spaces

Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit $$\varinjlim_{a\in J} K_a$$ for $J$ a directed set ...
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4 votes
0 answers
86 views

Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$

$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...
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1 vote
0 answers
66 views

Closed subgroups of totally disconnected Polish amenable groups

Let $G$ be a totally disconnected Polish topological group (e.g., a closed subgroup of the homeomorphism group of the Cantor set). If $G$ is amenable, is every closed subgroup of $G$ also amenable? ...
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9 votes
1 answer
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Group of isometries of Banach spaces a topological group?

Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$. Q: Is $\mathrm{Iso}(X)$ a topological group ...
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2 votes
0 answers
129 views

Closure of an amenable subgroup

Let $G$ be a topological group, and let $H < G$ be a countable subgroup that is amenable as a discrete group. Is the closure of $H$ an amenable topological group?
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3 votes
1 answer
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Fourier multipliers and transference on cyclic groups

It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
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11 votes
0 answers
232 views

Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?

Let $G$ be a finite group. It has been shown that: If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian. If the probability ...
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6 votes
1 answer
303 views

Is there a natural topology on the automorphism group of a topological group?

$\DeclareMathOperator\TAut{TAut}\DeclareMathOperator\Homeo{Homeo}$Let $G$ be a topological group, and let $\TAut(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the ...
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5 votes
0 answers
146 views

When exponential map is 1-1 from vector fields to diffeomorphisms

Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
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3 votes
3 answers
396 views

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 $\...
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5 votes
1 answer
223 views

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 $...
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5 votes
0 answers
146 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 ...
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2 votes
1 answer
196 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 ...
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  • 273
14 votes
3 answers
771 views

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 ...
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10 votes
0 answers
216 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 ...
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7 votes
2 answers
244 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 ...
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4 votes
1 answer
110 views

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 ...
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4 votes
2 answers
195 views

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 ...
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8 votes
1 answer
351 views

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 ...
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2 votes
1 answer
158 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 ...
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0 votes
0 answers
61 views

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?
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2 votes
1 answer
125 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 ...
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20 votes
0 answers
227 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 ...
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1 vote
0 answers
100 views

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}.$$ ...
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3 votes
0 answers
81 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\...
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2 votes
1 answer
139 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 ...
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5 votes
1 answer
143 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 $$\...
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3 votes
1 answer
135 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}...
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  • 2,657
0 votes
1 answer
117 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 ...
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  • 457
4 votes
1 answer
239 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 ...
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6 votes
1 answer
246 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 ...
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  • 2,657
6 votes
1 answer
233 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,...
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6 votes
2 answers
363 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 $U$. ...
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  • 61
14 votes
1 answer
369 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 \...
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  • 724
1 vote
1 answer
188 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 ...
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  • 59
5 votes
0 answers
145 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 ...
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6 votes
2 answers
219 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 ...
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7 votes
1 answer
189 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. ...
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