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2
votes
1answer
157 views

Tensor product of topological abelian groups with the reals

Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space. Now suppose that A is a topological abelian group (if necessary, we can assume it ...
6
votes
1answer
156 views

Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?

Let $G$ be a locally compact (second countable) group and let $$ G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}. $$ This is the kernel of the ...
3
votes
1answer
125 views

The finest countably generated free topological group so that $x^{m_{n}}_n\rightarrow 1$?

What is the finest free topological group $H$ with generators ${x_{1},x_{2},...}$ so that $x^{m_{n}}_n\rightarrow 1$ for all sequences $m_{1},m_{2},...$? Is $H \simeq K$, with $K$ the natural ...
1
vote
1answer
396 views

Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation. Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
4
votes
0answers
343 views

Completion of abelian topological groups

During some exercises I came upon the following questions that I cannot answer myself. Given a topological group $G$ we can build the completion using cauchy sequences and denote this completion by ...
2
votes
1answer
255 views

topological group that is connected and locally connected but not path-connected

Is there a ($\mathrm{T}_0$) topological group that is connected and locally connected but is not path-connected? This is a cross-post from MSE, since my question there was posted over three weeks ...
2
votes
1answer
83 views

Is every regular paratopological group completely regular?

This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open? Dusan
10
votes
3answers
661 views

To what extent has the Haar measure been generalized?

It is known that all locally compact groups, and therefore compact groups, have a left-invariant Haar measure which is unique up to scalar constant, also a right-invariant one. Is there a strictly ...
7
votes
2answers
322 views

Are the reduced group Von Neumann algebra/ Group $C^{\ast}$ algebra functorial in the case of LCH groups

Let $G$ be a LCH group and $\mu$ be its left Haar measure. Call $\lambda_G : G \to U(L_2(G,\mu))$ the left regular representation. We can define the reduced $C^{\ast}$ algebra and reduced Von Neumann ...
4
votes
1answer
102 views

Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$

Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form: $$ \| T_t : L_p(\mu) \to L_q(\mu)\| ...
7
votes
3answers
360 views

A Hausdorff abelian group with no character?

Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be ...
-1
votes
1answer
193 views

A characterization of the module function on a locally compact division ring

The same question was asked in Math StackExchange about 3 months ago. Since nobody has answered to it, I would like to post it here. References: Weil's Basic Number Theory(denoted by BNT). ...
4
votes
2answers
132 views

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 ...
12
votes
2answers
320 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 $a_{n} \in A$ ...
4
votes
1answer
116 views

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 ...
6
votes
1answer
502 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 ...
9
votes
0answers
177 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 $n$?
17
votes
0answers
502 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 ...
7
votes
2answers
532 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 ...
8
votes
2answers
253 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 ...
0
votes
2answers
146 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 ...
9
votes
4answers
1k views

Compact open topology

What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?
8
votes
0answers
240 views

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 ...
7
votes
1answer
328 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 ...
1
vote
0answers
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 ...
1
vote
2answers
170 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?
1
vote
0answers
173 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 ...
-4
votes
1answer
245 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?
0
votes
1answer
243 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 ...
18
votes
2answers
805 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 ...
3
votes
1answer
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: $G\times F\rightarrow F$, ...
0
votes
1answer
200 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 ...
1
vote
1answer
293 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 ...
1
vote
0answers
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 ...
1
vote
1answer
224 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 ...
15
votes
1answer
546 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 ...
1
vote
1answer
228 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.
6
votes
2answers
362 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 ...
5
votes
2answers
292 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 ...
1
vote
0answers
156 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 ...
11
votes
2answers
383 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 $\mathscr{R}$. ...
7
votes
2answers
318 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 ...
8
votes
5answers
2k 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 ...
2
votes
0answers
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 ...
4
votes
0answers
90 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 ...
1
vote
1answer
237 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} ...
7
votes
0answers
242 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 ...
5
votes
5answers
653 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 ...
6
votes
3answers
926 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 ...
6
votes
2answers
378 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 ...