The topological-groups tag has no wiki summary.

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### Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...

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### exotic compact group

Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...

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### What groups are Lie groups?

We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...

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### Criteria for topologically finitely generated profinite groups

Q1: Do we have a criterion which allows us to say when is a profinite group $G$ topologically finitely generated?
For example, if $G$ is topologically finitely generated then, for a fixed integer ...

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### Topologically split extensions of topological groups

Let $1 \to N \to G \to H \to 1$ be a short exact sequence of topological groups. Such an exact sequence is said to be topologically split if $G$ is $N \times H$ as a
topological space.
Can someone ...

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### On closed totally disconnected subgroups of connected real Lie groups

So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...

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### Picking a representative in a continuous way

I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup.
...

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### Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...

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### Examples of totally disconnected, locally compact non-sigma-compact groups

I am looking for examples of totally disconnected, locally compact groups, which are not sigma-compact. For a start any such an example would do, so that I can a feeling for those groups and how to ...

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### Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of contiunous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...

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### Which principlal bundles are locally trivial?

If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial.
In the wikipedia article on ...

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### When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...

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### Epimorphisms have dense range in TopHausGrp?

Consider the category of Topological Groups with continuous homomorphisms. Then a continuous homomorphism $f:G\rightarrow H$ with dense range is an epimorphism. Is the converse true? If not, what ...

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### Non-isomorphic two-transitive permutation groups with isomorphic point stabilizers

The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ ...

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### Why is the dual of a torus the same as its fundamental group?

The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...

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### Locally compact abelian groups

First, some preliminaries:
Define an "LCA group" to be a locally compact Hausdorff abelian topological group.
Define "smooth manifold" in a way that requires Hausdorffness, but not connectedness or ...

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### Topological Groups and Families of Pseudometrics

The topology on a topological group is generated by a family of pseudometrics. The only proof I know passes through uniform spaces (by which I mean the entourage definition): A topological group has ...

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### Irreducible unitary representations of locally compact groups

Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know
that $\mu$ is unique up to a scalar in $\mathbf{R}_{>0}$. I don't know so much about unitary representations of ...

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### Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...

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### When is the group of homeomorphisms of a compact space locally compact?

When is the group of homeomorphisms of
a compact space locally compact?
I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate ...

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### index of a closed subgroup of a profinite group

In the book "profinite groups, arithmetic, and geometry" of Shatz, the index $(G:H)$ of a closed subgroup $H$ of a profinite group $G$ is defined to be the supernatural number $lcm\big((G/U):(H/(H\cap ...

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### Compact Topological Group Properties [closed]

I feel I want to understand it better. I know that for every cover there's a finite subcover but what can you say about it's group properties?
I'm stuck on this homework problem where we were asked:
...

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### Are there non-reflexive abelian topological groups isomorphic to their second dual?

I posted the following question in a comment at
Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...

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### Closed connected additive subgroups of the Hilbert space

It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...

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### orbits in locally compact group

As everyone knows if $x\in S^1$, then the set $\{ x^n \}$ is either finite or dense. Under which condition is true for any other locally compact group, i.e if $G$ is a locally compact group, and $x\in ...

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### Does homeomorphic and isomorphic always imply homeomorphically isomorphic?

Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that
$(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$.
Does it follow that $(G,\cdot,T)$ and ...

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### Is there a group homeomorphic to but not homeomorphically isomorphic to the circle group?

Let $Circ$ be the topological group
$(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot , \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z ...

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### Fundamental groups of topological groups.

Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is:
If $G$ is a ...

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### Which groups can be recovered from their unitary dual?

Note: in this post, every topological group under consideration is assumed to be Hausdorff.
Given a locally compact abelian group, one can construct its dual group, i.e. its group of (unitary) ...

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### existence of charaterization of amenable groups by complementation?

Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...

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### How to define the quotient of a measure which is invariant under group action?

I am a physicist, and I have the following problem. Consider a locally compact group G acting over a measure space $(X, {\cal B}, \mu)$, and assume that $\mu$ is G-invariant. My problem is how to ...

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### Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...

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### When does a closed inclusion induce a closed inclusion on free topological groups?

Suppose $A$ is a closed subspace of a Tychonoff space $X$. Does the inclusion $i:A\hookrightarrow X$ induce a closed embedding of topological groups on the free (Markov -the unbased version) ...

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### Examples of certain locally compact totally disconnected groups

To find a counterexample disproving a generalization of a theorem in the theory of scale functions on locally compact totally disconnected groups, initiated by George Willis, I am looking for a group ...

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### Dense cyclic subgroup

Does anyone know a continuous group (not necessarily locally compact) with dense cyclic subgroup other than a torus?

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### Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology

It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff ...

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### Sequential topological vector spaces

Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree ...

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### Are there locally compact groups which have no compact open subgroups and no discrete infinite cyclic subgroups?

The posting of this question was suggested by Yemon Choi: see Discrete cyclic subgroup.. The question is not mine; it's just a rephrasing of Discrete cyclic subgroup.
EDIT 4. This post claims that ...

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### Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions

The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...

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### Why does the generic pair generate a dense subgroup of a connected compact Polish group? (cf. Schreier and Ulam)

A result of Schreier and Ulam from their 1935 paper "Sur le nombre des g$\acute{\textrm{e}}$n$\acute{\textrm{e}}$rateurs d'un groupe topologique compact et connexe" says that if $G$ is a connected ...

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### What is the situation with Hilbert's Fifth Problem ?

The common knowledge in this regard seems to be that it was completely solved in the 1950s by a few Americans. About a decade ago, Olver contested that in one of his books. Recently, Palais wrote ...

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### Example of a quasitopological group with discontinuous power map

A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion ...

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### Discrete cyclic subgroup.

Let T is a hausdorff group topology and (G,T) is locally compact abelian group.If (G,T) has no open compact subgroups then can we say G has an infinite discrete cyclic subgroup?

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### Group homomorphisms and maps between function spaces

Let G and H be locally compact groups, and let $\theta:G\rightarrow H$ be a continuous group homomorphism. This induces a *-homomorphism $\pi:C^b(H) \rightarrow C^b(G)$ between the spaces of bounded ...

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### When are the homology and cohomology Hopf algebras of topological groups equal?

Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology ...

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### Fundamental group as topological group

Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...

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### Is the ability to define Haar measure the main (or only) reason to consider locally compact topological groups? [closed]

Because I haven't seen locally compact topological groups used for anything except Fourier analysis.

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### Which compact groups have finitely many irreducible representations of each dimension?

If my understanding is correct, this is true of sufficiently nice nonabelian Lie groups (see Ben Webster's answer below), and any finite group. On the other hand, this is false for any infinite ...

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### Must a locally compact group be Hausdorff in order to possess a Haar measure?

Does the existence of (left) Haar measure on a locally compact topological group require that the group be Hausdorff?

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### Haar Measure on a Quotient [closed]

Suppose you have a locally compact group G with a discrete subgroup H. Of course G has a unique (up to scalar) Haar measure, but it seems that G/H has and induced Haar measure as well.
How does ...