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2
votes
1answer
90 views

The evaluation fibration of a transitive, effective topological group action

Does anybody know a reference to the following fact? If $G$ is a topological group acting transitively and effectively on a space $X$, then the evaluation map $G \rightarrow X$, $g \mapsto g \cdot ...
0
votes
1answer
57 views

A group topology which commutes with closed subgroups

For a topological group $(G,\mathcal T)$ and a subgroup $H\le G$, we say $\mathcal T$ and $H$ are permutable if for every neighborhood $U$ of $1$, there is a neighborhood $V$ of $1$ with $VH\subseteq ...
4
votes
2answers
202 views

Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
0
votes
1answer
62 views

Bohr compactification and “discretization”

Let $G$ is a compact group. We can form the Pontriagin dual $\widehat{G}$ of $G$: it is then discrete space. One can consider the Bohr compactification $b\widehat{G}$ of $\widehat{G}$ which is compact ...
48
votes
1answer
3k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
1
vote
0answers
39 views

Just-not-nilpotent-by-compact quotient of a locally compact group

It is known (and not complicated to prove) that for a finitely generated not virtually nilpotent group $G$, we can pass to a quotient $G/N$ of $G$, such that the quotient is just not virtually ...
3
votes
2answers
274 views

Exotic group topologies on the affine group $ax+b$

Let $G = \{(x; y) : x \in \mathbb{R}, y > 0\}$. With $(x, y)(u, v) = (x + yu, yv)$, $G$ is a group. If we topologize $G$ as a subset of $\mathbb{R}^2$, it is known that $G$ is a locally compact ...
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 $G$ ...
17
votes
4answers
2k views

Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...
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 ...
7
votes
2answers
506 views

Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
5
votes
1answer
256 views

Is the unitary group of a pre Hilbert space contractible?

I already posted my question on mathstackexchange For a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for ...
0
votes
1answer
95 views

Borel subsets of Polish groups

Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...
5
votes
1answer
109 views

Why is it possible to normalize the Haar measure on the quotient?

I just asked a question which is related to the one I'm about to ask, but I realized my question can be reduced to the following: let $G$ be a locally compact abelian group with Haar measure $\mu$, ...
0
votes
0answers
36 views

The use of Haar measure in the Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
-2
votes
1answer
84 views

Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?

Is there somone help me to show that if this problem have positive Answer : Problem :Can every non-discrete topological group G be algebraically gen- erated by a nowhere dense subset ? Thank ...
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 ...
1
vote
1answer
77 views

Number of continuous characters on an infinite Hausdorff precompact abelian group with exponent $p$

Let $(G,\mathcal T)$ be an infinite Hausdorff precompact abelian topological group and let $G$ have exponent $p$ where $p$ is a prime number. Can it be proved that there are at least $p+1$ continuous ...
5
votes
0answers
85 views

Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre. Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
0
votes
0answers
13 views

abelian subgroups of the automorphism group of a totally disconnected LCA group

I am interested in the following question. Suppose that $A$ and $B$ are LCA groups and $B$ acts continuously on $A$ by topological automorphisms. If $f$ is a Schwartz function on $A$, then we want to ...
6
votes
1answer
108 views

Continuity of conjugation actions of Polish groups

Let $G$ and $H$ be Polish groups and let $\psi: G \rightarrow H$ be a continuous injective homomorphism such that $\psi(G)$ is normal in $H$. Then $H$ acts on $G$ by conjugation via $\psi$, in other ...
11
votes
3answers
876 views

Are measurable homomorphisms $ (\Bbb{C},+) \to (\Bbb{C},+) $ or $ (\Bbb{C},+) \to (\Bbb{C},*) $ continuous, and do they admit an explicit description?

I am interested in generalizations of the following fact (known as automatic continuity, as pointed out below). I am especially looking for references to papers dating back to 1920’s. I feel that ...
1
vote
2answers
124 views

query about quasi-simple algebraic groups over local fields

Suppose that $G$ is an absolutely quasi-simple algebraic group over a non-archimedean local field $k$ (of either zero or positive characteristic). Is it known whether or not it is necessarily the case ...
4
votes
1answer
186 views

Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...
11
votes
4answers
828 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 ...
4
votes
0answers
68 views

Topological systems of imprimitivity

Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense. Here is an attempt to define ...
3
votes
4answers
250 views

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite ...
1
vote
3answers
159 views

Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?

In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
2
votes
1answer
88 views

Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously

My question will rely on a clarification of a proof, which I simply don't understand. Let us denote by $X$ a pseudo-riemannian symmetric space and define $$ Z_{\mathrm{Iso}\left(X\right)}G(X) = \{\, ...
1
vote
1answer
104 views

Continuity of inversion and composition in certain topological groups

For $k\in\Bbb{N}$ ($k\geqslant 1$) and $\alpha\in]0,1]$, let $H_{k,\alpha}([0,1])$ be the group of orientation preserving $C^{k,\alpha}$ diffeomorphisms of the closed unit interval $[0,1]$. We furnish ...
0
votes
1answer
95 views

Powers in compact coset spaces

Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence (edit: or net) of ...
3
votes
2answers
146 views

Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$. Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...
16
votes
1answer
726 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 ...
2
votes
1answer
92 views

Coarsely trivial Borel cross section for $G\to G/N$

Let $G$ be a locally compact group, and let $N$ be a closed, normal subgroup, and let $\pi\colon G\to G/N$ be the quotient homomorphism. It is known that there exists a Borel cross section, i.e., a ...
3
votes
1answer
399 views

Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space ...
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 $a_{n} \in A$ ...
1
vote
1answer
92 views

A countable tight topological group where every countable subset is metrizable

I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example. ...
1
vote
1answer
73 views

Structure of locally compact non discrete topological division algebras without the use of Haar measure

There is a well-known structure theorem for locally compact non discrete topological division algebras, see here http://math.stackexchange.com/q/1160086/187521 (I repost it here because I think it ...
4
votes
1answer
244 views

Can an abelian group have a minimal group topology?

In the abstract of this paper, it is said that a minimal group topology on an abelian group is not Hausdorff. Suppose $G$ is an abelian group and $\mathcal T$ is a minimal group topology on $G$ and ...
3
votes
2answers
233 views

Exact sequence of groups to exact sequence of sheaves

Disclaimer: This is a cross-listing of a math.stackexchange post. While not research level, after a week of no response, I figured I would ask it here. For a topological group $G$ and a topological ...
13
votes
1answer
281 views

Anything between vector bundles and sphere bundles?

There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical ...
0
votes
0answers
47 views

Is the core of an atom in lattice of group topologies a coatom?

Let $(G,\mathcal T)$ be an abelian topological group such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we have $\mathcal S = \mathcal T$. Let ...
1
vote
1answer
107 views

A Hausdorff atom in lattice of group topologies

Do you have an example of an infinite Hausdorff nonabelian topological group $(G,\mathcal T)$ such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we ...
0
votes
0answers
42 views

Definition of submaximal (group) topology

By [3] (or [2]), a submaximal group topology on a group is the infimum of all maximal group topologies on it. By [4], a submaximal topological space is one with all dense subsets open. By [1], a ...
2
votes
1answer
133 views

Eigenfunction of ergodic skew product fixed by commutator?

Background: Let $(Y, \mathcal{B},\mu,T)$ be an ergodic probability system and let $G$ be a compact metrizable group with compact subgroup $H$. Given a measurable map $\rho:Y \to G$. We may define the ...
2
votes
1answer
111 views

Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?

The following problem is a stumbling block in a research project that I am working on: Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it ...
28
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 ...
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 ...
10
votes
0answers
196 views

Is the quotient map of the action of homeomorphisms on embeddings well-behaved?

It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the ...
1
vote
1answer
90 views

Is every closed subgroup of dual group an annihilator?

Let $G$ be a LCA group. It is well-known that to every closed subgroup $H$ of $G$ correspond a closed subgroup in the dual group $\widehat{G}$, namely the annihilator of $H$. My question is this : is ...