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10
votes
4answers
886 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 ...
10
votes
1answer
415 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 ...
11
votes
1answer
692 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 $\mathbb{F}_8$ ...
9
votes
3answers
935 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 ...
7
votes
2answers
1k views

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 ...
3
votes
1answer
349 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 ...
2
votes
2answers
974 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 ...
12
votes
1answer
689 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 ...
1
vote
2answers
694 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 ...
7
votes
1answer
480 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 ...
0
votes
1answer
775 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: ...
8
votes
0answers
391 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 ...
13
votes
0answers
471 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 ...
5
votes
4answers
505 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 ...
36
votes
6answers
3k 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 ...
-1
votes
2answers
640 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 ...
15
votes
4answers
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 ...
7
votes
1answer
618 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) ...
4
votes
1answer
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 ...
4
votes
2answers
924 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 ...
6
votes
0answers
397 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 ...
4
votes
0answers
182 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) ...
5
votes
2answers
854 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 ...
1
vote
4answers
417 views

Dense cyclic subgroup

Does anyone know a continuous group (not necessarily locally compact) with dense cyclic subgroup other than a torus?
8
votes
3answers
514 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 ...
2
votes
3answers
797 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 ...
5
votes
3answers
909 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 ...
7
votes
1answer
481 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 ...
3
votes
1answer
245 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 ...
-4
votes
4answers
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 ...
4
votes
1answer
286 views

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 ...
1
vote
1answer
344 views

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?
8
votes
2answers
641 views

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 ...
3
votes
3answers
928 views

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 ...
24
votes
4answers
3k views

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 ...
4
votes
3answers
660 views

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.
8
votes
2answers
621 views

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 ...
2
votes
1answer
484 views

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?
5
votes
4answers
3k views

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 ...
25
votes
7answers
3k views

Why are free groups residually finite?

Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a ...
6
votes
1answer
300 views

Homomorphisms of Topological Groups which are Automatically Fiber Bundles?

Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...
10
votes
3answers
670 views

How do you recover the structure of the upper half plane from its description as a coset space?

This is maybe a dumb question. $SL_2(\mathbb{R})$ has a natural action on the upper half plane $\mathbb{H}$ which is transitive with stabilizer isomorphic to $SO_2(\mathbb{R})$. For this reason, ...
9
votes
2answers
533 views

Is there a non-trivial topological group structure of $\mathbb{Z}$?

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
1
vote
2answers
329 views

Smoothness as a topological property

Motivation: Let $G$ be an $\ell$-group (locally profinite group). A map $G\to \mathbb{C}$ is called smooth provided that it is continuous as a map $$G\to \mathbb{C}_{discrete}.$$This gives us the ...
11
votes
1answer
800 views

Topological HNN extensions

First, let me recall what an abstract HNN extension is. Let $G$ be an abstract group, $A, B < G$ be subgroups of $G$ and $\phi : A \to B$ be an isomorphisms. Then there is a group $H$ and an ...
1
vote
1answer
413 views

Infinite products of topological groups

While studying for a topological groups course, I wondered if we could define the product of uncountably many topological groups such that the product is still a topological group. That is: let $G_i$ ...
13
votes
7answers
2k views

Haar measure on a quotient, References for.

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz). Right from the very beginning, Weil uses the ...
13
votes
3answers
806 views

Countable subgroups of compact groups

What is known about countable subgroups of compact groups? More precisely, what countable groups can be embedded into compact groups (I mean just an injective homomorphism, I don't consider any ...
7
votes
2answers
405 views

Potential connected non-Lie subgroup

This painful question is inspired by the question "non-Lie subgroups" . Let R denote the real numbers. Let f be an discontinuous additive map from R --> R. Is it possible that the graph of f, inside ...