# Tagged Questions

The topological-groups tag has no wiki summary.

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### A restatement, in terms of the semi-group product of the left-invariant completion of a Polish group, of http://mathoverflow.net/questions/71389

This is a re-statement, of sorts, of Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?, so far unanswered.
Let $G$ be a ...

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922 views

### Finite dimensional vector spaces over a complete but not-necessarily-valued field

I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...

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248 views

### When does a LCA group not contain a (closed) infinite cyclic subgroup?

If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...

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823 views

### Haar measure for large locally compact groups

In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a ...

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362 views

### Topology on extensions of topological groups

Let $G$ and $H$ be two topological groups and let $\mathcal{E}:0 \to G \to E \to H \to 0$ be an extension of abstract groups.
Is there a way to introduce a topology on $E$ such that $\mathcal{E}$ ...

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vote

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347 views

### Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$

What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups ...

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161 views

### Shrinking Group Actions

This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a ...

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**1**answer

485 views

### On homeomorphic compact connected topological groups

I wish to thank Professor Claudio Gorodski for his very helpful
answers to my question on the webcite:
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie ...

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vote

**1**answer

276 views

### Have a general mean of an amenable group a kind of Fubini's property ?

Recently in my investigations I faced a problem related to amenable groups.
I have no idea if my question is suitable for this site, but after ask some collegues in my department I decided to try to ...

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167 views

### 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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182 views

### 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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882 views

### 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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651 views

### 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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**3**answers

549 views

### 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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**3**answers

650 views

### 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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**1**answer

308 views

### 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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848 views

### 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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**2**answers

616 views

### 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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**4**answers

1k views

### 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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**1**answer

981 views

### 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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821 views

### 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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406 views

### 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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673 views

### 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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893 views

### 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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**1**answer

337 views

### 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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926 views

### 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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**1**answer

639 views

### 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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665 views

### 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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448 views

### 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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749 views

### 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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386 views

### 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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466 views

### 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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493 views

### 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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2k views

### 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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628 views

### 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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2k views

### 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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**1**answer

594 views

### 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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**1**answer

205 views

### 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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815 views

### 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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396 views

### 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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**0**answers

179 views

### 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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827 views

### 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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406 views

### 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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508 views

### 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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756 views

### 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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820 views

### 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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465 views

### 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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**1**answer

240 views

### 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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3k views

### 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 ...