The topological-groups tag has no wiki summary.

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### A theorem of Markov about completely regular spaces and topological groups

In Pontriaguin's classic book Grupos continuos (in English Continuous Groups), says that A. Markov proved that:
There are topological groups that are not normal.
Furthermore, he says it is ...

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316 views

### If G is a sequential topological group, must G x G be sequential?

Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$
so that $a_{n} \in A$ ...

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### Layman question: A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is unlikely a research level question... one that would be answered in a blink of an eye, rather...it is an (early) exercise from the book "Analytic Pro-p groups". But since no reply was received ...

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**1**answer

488 views

### Naive question on adelic groups

The ever-reliable Wikipedia says:
... an adelic algebraic group is a semitopological group defined by...
No more details are given, and I was wondering if the multiplication only being ...

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175 views

### Contractible topological groups

Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?

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492 views

### Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$:
$A' = aA$,
$B' = bB$.
Suppose it is known that ...

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487 views

### Conditions for a topological group to be a Lie group

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157):
Let $G$ be a locally compact ...

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238 views

### Why is TopGrp the category of topological groups and continous homomorphisms protomodular?

Why is TopGrp the category of topological groups and continous homomorphisms protomodular? I know it is, and I have several indirect proofs, but am not able to prove this directly by showing that the ...

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**1**answer

110 views

### Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?

Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
clearly every closed neighborhood ...

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### Compact open topology

What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?

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### Which topological spaces are coset spaces of locally compact groups?

What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ?
Such a space $X=G/H$ necessarily ...

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**1**answer

318 views

### Are finite index subgroups of inertia closed?

Let $K$ be a finite extension of the $p$-adic numbers. $G_K$ be its absolute Galois group and $I_K$ the inertia subgroup. Are finite index subgroups of $I_K$ closed in its profinite topology?
By a ...

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41 views

### a free topological product of topological semigroups?

Much work has be done on describing the topology of free products of topological groups (Graev, Morris, Katz, etc). Could anybody hint me any results on free topological products of topological ...

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168 views

### Cross section for closed Lie subgroup in a Lie group

Let $G$ be a Lie group and $H$ a closed Lie subgroup. Is there an explicit way to construct a local cross section of $H$ in $G$ so that $\pi: G\to G/H$ is a fiber bundle?

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172 views

### From positive definite function to Følner sequence --— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...

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**1**answer

239 views

### 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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237 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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785 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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204 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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**1**answer

188 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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286 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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196 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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211 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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529 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

225 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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359 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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288 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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152 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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378 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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### 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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### 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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123 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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### 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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236 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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232 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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644 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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### $\pi_1$ 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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### 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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371 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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556 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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### $2$-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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368 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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214 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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### 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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### 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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368 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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### 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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### 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 ...