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0
votes
1answer
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 ...
8
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
215 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 ...
6
votes
1answer
299 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
155 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
170 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
232 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
234 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 ...
17
votes
2answers
741 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
203 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
177 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
270 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
195 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
202 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
515 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
224 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
345 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
278 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
144 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
369 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
298 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
1k 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
122 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
88 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 ...
0
votes
1answer
232 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
229 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
626 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
915 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
377 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 ...
2
votes
0answers
122 views

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 ...
0
votes
1answer
332 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 ...
2
votes
3answers
533 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 ...
15
votes
2answers
769 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 ...
1
vote
1answer
365 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 ...
6
votes
1answer
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 $v:K^{\times}\to\Gamma$ ...
1
vote
0answers
420 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 ...
2
votes
1answer
634 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 ...
3
votes
1answer
365 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 ...
8
votes
0answers
255 views

'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 ...
0
votes
1answer
573 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^* ...
5
votes
0answers
203 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 ...
6
votes
3answers
583 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 ...
7
votes
4answers
708 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 ...
6
votes
2answers
976 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 ...
12
votes
1answer
941 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 ...
0
votes
1answer
547 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 ...
0
votes
0answers
110 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) ...
26
votes
3answers
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 ...
3
votes
0answers
209 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 ...