The topological-groups tag has no wiki summary.

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**1**answer

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### Is the product of closed subgroups in a locally compact group locally compact? [closed]

Let $A$ and $H$ be closed subgroups of a $\sigma$-compact locally compact group $G$. Assume further that $A$ is abelian. Is the group $AH$ locally compact subgroup in the subspace topology?

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**1**answer

201 views

### Is the image of discrete set under an open map discrete?

Let $G$ and $H$ be locally compact totally disconnected abelian groups, and $f:G\rightarrow H$ a surjective open map. Let $Y\subseteq G$ be a discrete subgroup in the subspace topology. Is it true ...

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**2**answers

620 views

### Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?

The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...

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votes

**1**answer

193 views

### a free topological group as a topological module

Let $G$ be a topological group, and let $F$ be the Markov free topological group over $G$. We define an action of $G$ on $F$ as follows: $G\times F\rightarrow F$, ...

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votes

**1**answer

129 views

### Strongly Complete Profinite Groups.

I've been reading about profinite groups and have encountered the notion of strong completeness. I.e. that a profinite group $G$ is strongly complete if it is isomorphic to it's profinite completion ...

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vote

**1**answer

241 views

### Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...

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**0**answers

186 views

### does s.e.s 0->A->B->C->0 of profinite groups imply C=B/A and A<B topologically?

Assume $A, B, C$ are profinite groups and $0\to A\to B\to C\to 0$ is an exact sequence of continuous maps. Which of the following assertions follows?:
(i) the subspace-topology induced on $A$ via ...

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**1**answer

175 views

### The Chabauty space

Hello,
Let $G$ denote a locally compact group and $S(G)$ the chabauty space of $G$, that is the set of closed subgroups of $G$ equiped with the chabauty topology, it is a compact space.
My question ...

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**1**answer

443 views

### Is a reductive adelic group a Type I group?

I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...

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**1**answer

217 views

### Recommend a book about compact subgroups

Hi, could you please recommend me some books/articles where I could find information about compact subgroups of metric topological compact (abelian) groups? Thanks in advance for any help.

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**2**answers

304 views

### The integers as a sequential but non-first countable topological group

Completely unaware of the Bohr topology, I recently asked whether or not there was a Hausdorff group topology on the integers $\mathbb{Z}$ which made the group fail to be first countable. For me, this ...

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**1**answer

203 views

### Hausdorff group topologies on finitely generated groups

Suppose $G$ is a finitely generated Hausdorff topological group. Must $G$ be first countable (or perhaps a sequential space)? What if we restrict to the abelian case?
I wonder if this is even true ...

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**0**answers

123 views

### Intersection of cocompact closed normal subgroups

Let $G$ be a locally compact Hausdorff topological group.
Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology.
Note ...

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**2**answers

334 views

### Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]

And what else can be said, if so?
(Original math.SE post)
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. ...

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**2**answers

280 views

### non-artificial examples of non-smooth and non-admissible representations of GL_2

Let $F$ be a finite degree extension over $\mathbf{Q}_p$ and consider the locally profinite group $G:=GL_2(\mathbf{Q}_p)$.
P1: Give an interesting example (non-artificial one, i.e., one that arises ...

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**5**answers

996 views

### What is a good book on topological groups?

I am looking for a good book on Topological Groups. I have read Pontryagin myself, and I looked some other in the library but they all seem to go in length into some esoteric topics.
I would love ...

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**0**answers

121 views

### Sigma-compactness in Furstenberg paper

I've been reading the classical Furstenberg paper "The structure of distal flows", where the author claims he is working with an arbitrary locally compact group $T$. Nevertheless, the proof of Lemma ...

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**0**answers

85 views

### A dynamical property of automorphisms of a locally compact group

Let $G$ be a Hausdorff locally compact group and let $\alpha$ be an automorphism of $G$. Say $\alpha$ is (forwards) topologically recurrent if for all $g \in G$ and all neighbourhoods $O$ of $g$, the ...

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**1**answer

226 views

### Harmonic Analysis [closed]

Let $G$ be a locally compact group, $H$ be a closed subgroup and $N$ be a normal subgroup of $G$ such that $H\subseteq N$. How can we get $$\int_{G/H} ...

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208 views

### Strange normal subgroups of profinite groups

I am looking for an example of the following situation:
$G$ is an infinite profinite group, with a dense normal subgroup $N$. However $N$ does not contain any non-trivial closed normal subgroup of ...

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**5**answers

548 views

### Commutator of algebraic subgroups is connected

Let $G$ be an algebraic group over an
algebraically closed field. If $H$ and
$K$ are closed subgroups and one of
them is connected, then their
commutator $[H,K]$ is also connected.
Is ...

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**3**answers

877 views

### $\pi_1$ Sequence of Topological Groups

Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...

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337 views

### How big $|Aut(M)|$ can be, given $|\partial Aut(M)|$?

My apologies: There were a couple of typos in the original question. Hope I got them all.
Let $\kappa$ be an uncountable cardinal of cofinality $\omega$ and $M$ a model of size $\kappa$. We equip ...

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### Almost conjugation-invariant neighborhoods of units in locally compact groups

Let $G$ be a locally compact topological group with unity $e$ and left Haar measure $m$. Let also $g\in G$ be a given element and $U$ a neighborhood (of compact closure) of $e$.
I am interested to ...

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**1**answer

278 views

### About the closure of a commutative subgroup of a topological group.

Let $H$ a topological subgroups of a topological group $G$, and $H'$ the closure of $H$ as topological subspace.
Are classic results the if $H'$ is a topological subgroup, and that it is normal if ...

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415 views

### When is a topological group Hausdorff (separated)?

Does someone knows a good reference for the following result?
"A topological group is Hausdorff if and only if the identity is closed."
I have seen proofs in lecture notes of courses on the web, but ...

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739 views

### $2$-categorical structure in Grothendieck's Galois Theory

Grothendieck's Galois Theory, as developed in SGA I, V.4, or very gently in Lenstra's notes, establishes an equivalence between profinite groups and Galois categories. We can put this into the ...

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**1**answer

335 views

### Finiteness theorems for profinite groups

Let $G$ be a profinite group which fits in the following short exact sequence:
$$
1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1
$$
Assume that $N$ is a pro-$p$ group and that $K$ is ...

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**1**answer

195 views

### When is a valued field second-countable?

Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial).
The valuation $v:K^{\times}\to\Gamma$ ...

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355 views

### Why groups that admit Folner Sequences are amenable

I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...

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**1**answer

624 views

### Where is the error in this argument?

Let $G$ be a locally compact Hausdorff group.
It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T ...

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353 views

### Are the categories of representations of G and C*(G) isomorphic?

Let G be a locally compact Hausdorff group, and C*(G) the full group C* algebra.
I found in some books that "representation theory of both is the same". Can this be expressed as "the categories are ...

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### 'Infinitesimal' elements of a topological group

Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close ...

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470 views

### Proof that the Pontryagin dual of a topological group is a topological group

I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* ...

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189 views

### Examples of a non-Hopfian phenomenon in group theory

I am interested in examples of the following property, where $G$ is a non-discrete locally compact topological group:
(*) The open normal subgroups of $G$ have trivial intersection, but $G$ has an ...

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581 views

### Does every compact Hausdorff ring admit a decomposition into primitive idempotents?

Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in ...

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644 views

### Measures on general topological groups

I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...

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**2**answers

787 views

### Two Definitions of “Character” of topological groups

When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
Let $G$ be a topological group. A character of $G$ is a ...

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**1**answer

880 views

### (Closures of sets of) operations in topological groups.

Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$.
Is there a ...

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**1**answer

508 views

### No non-trivial homomorphism to a group

Here is a question I posted some months ago in Math.SE, and t.b. mentioned to the following question by Florent MARTIN which is somehow related to my question;
Let $G$ be a compact Hausdorff ...

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107 views

### A continuous map from a T2 & compact space to a uniform space is uniformly continuous.

Can you recommend some literature that give a proof of this statement, and who allegedly prove it first?
BTW, is there any use of uniform spaces or topological spaces in mathematical (or theoretical) ...

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2k views

### morphism from a compact group to Z ?

I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism
$\phi : G \to \mathbb{Z}$ (without assuming any topological ...

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205 views

### When Aut(M) preserves a linear order?

I have a general-type question:
Suppose $M$ is a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an ...

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250 views

### classifying continuous functions on the $ax+b$ subgroup of $GL(2,\mathbb R)$

Let $G$ be 'ax+b' topological group i.e subgroup of $GL(2,\mathbb R)$ containing $2\times 2 $ matrices of type $\{\left(\begin{matrix} a&b\\\0&1\end{matrix}\right): a\neq 0\; a,b\in \mathbb ...

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128 views

### Categories with canonical factorizations into products satisfying two particular properties

An old splitting theorem for (Hausdorff) locally compact abelian (LCA) groups says that any LCA group $L$ is isomorphic to a direct product of $\mathbb{R}^n$ and $L_1$, where $L_1$ contains a ...

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218 views

### Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...

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567 views

### Does ZF prove that topological groups are completely regular?

Let $\mathbf{G} = \langle G,\cdot,\mathcal{T}\;\rangle$ be a topological group. Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$.
Assume $\{\mathbf{e}\}$ is closed in $\langle ...

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**1**answer

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### Complete metrics in locally compact topological groups

Hello,
I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group

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**2**answers

304 views

### Basic question on minimal flows

I know that minimal flows are actions for which no proper closed invariant subsets exist, but I am unclear how to understand this concept.
If a coset flow on a quotient space Gamma/S is ergodic, ...

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**1**answer

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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 ...