**5**

votes

**1**answer

295 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

**1**answer

130 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

**1**answer

151 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 $\mu$, ...

**7**

votes

**2**answers

568 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 ...

**-2**

votes

**1**answer

95 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 ...

**9**

votes

**1**answer

272 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 ...

**6**

votes

**1**answer

122 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 ...

**4**

votes

**1**answer

252 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 ...

**4**

votes

**0**answers

74 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

**4**answers

287 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 $...

**2**

votes

**1**answer

114 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

**1**answer

79 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 ...

**4**

votes

**2**answers

176 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. ...

**3**

votes

**1**answer

105 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 ...

**1**

vote

**1**answer

107 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

**1**answer

104 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

**2**answers

304 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 ...

**4**

votes

**1**answer

255 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 ...

**1**

vote

**3**answers

163 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 ...

**14**

votes

**1**answer

379 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 ...

**1**

vote

**1**answer

112 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 ...

**3**

votes

**1**answer

125 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 ...

**10**

votes

**0**answers

213 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 $C^\...

**1**

vote

**1**answer

147 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 ...

**0**

votes

**1**answer

101 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 ...

**2**

votes

**0**answers

101 views

### Selecting dense diagonals in $\Bbb T^2$

Let $p$ be a prime number and let $G=\bigcup_{n\in \Bbb N}\{\exp(k\frac{2\pi i}{p^n})\mid k\in \Bbb Z\}$ be a Prüfer group. For homomorphisms $f,g:G\to G$ let $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. ...

**1**

vote

**0**answers

98 views

### Hausdorff topologies on Q

Is there any description known of the Hausdorff topologies on $\mathbb{Q}$ compatible with the group operations?

**0**

votes

**0**answers

80 views

### Parallel topologies on a Prüfer group with the trivial group topology as the only group topology contained in both

Let $p$ be a prime number. A homomorphism $f:\Bbb Z_{p^\infty}\to \Bbb T$ induces a group topology $\mathcal T_f$ on $\Bbb Z_{p^\infty}$ with a base of neighborhoods $\mathcal N_f$ of $0$.
Are there ...

**2**

votes

**1**answer

250 views

### A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$

Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...

**1**

vote

**0**answers

29 views

### countably-infinite-index subgroup of a strongly complete profinite group

If $H$ is a strongly complete profinite group and $K$ is a dense countably-infinite-index subgroup, then I'm assuming a proper finite-index subgroup of $K$ could still be dense in $H$. Is there any ...

**2**

votes

**1**answer

146 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 ...

**1**

vote

**1**answer

75 views

### locally topologically finitely generated t.d.l.c. group

I am trying to find an example of the following situation.
$G$ is a t.d.l.c. (totally disconnected locally compact) $\sigma$-compact topological group in which every compact open subgroup is ...

**4**

votes

**1**answer

158 views

### Are convolution algebras ever “topologically noetherian”?

For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means ...

**1**

vote

**0**answers

75 views

### Topologically finitely generated residually finite group

Suppose that $G$ is a topologically finitely generated profinite group,
and let $H$ be a subgroup of countably infinite index. Is $H$ necessarily topologically finitely generated with the subspace ...

**1**

vote

**1**answer

98 views

### countably-infinite-index subgroup of a finitely generated profinite group

Suppose that $G$ is a profinite group with the property that every open compact subgroup is topologically finitely generated and just infinite. Suppose that $H$ is a commensurated subgroup of $G$ with ...

**2**

votes

**2**answers

304 views

### If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer ...

**4**

votes

**1**answer

141 views

### If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...

**7**

votes

**1**answer

394 views

### discrete group cohomology vs continuous group cohomology for profinite groups

Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map $...

**6**

votes

**0**answers

259 views

### Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...

**2**

votes

**1**answer

118 views

### Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?

**3**

votes

**0**answers

136 views

### Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$

Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$.
Are there group topologies $\mathcal A$ ...

**3**

votes

**1**answer

151 views

### algebraic groups over non-archimedean local fields acting on buildings

I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and ...

**1**

vote

**2**answers

147 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 ...

**13**

votes

**1**answer

1k views

### Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
[1] ...

**7**

votes

**1**answer

440 views

### Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$

Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function
$$f:G\times G\to G$$
$$f(x,y)=xy^{-1}$$
is continuous at $(1,1)$?

**4**

votes

**1**answer

187 views

### Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
The question is inspired by and related to this ...

**29**

votes

**1**answer

1k views

### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$...

**0**

votes

**1**answer

77 views

### Parallel group topologies on Prüfer groups

Let $p$ be a prome number. Are there group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that
$$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$

**6**

votes

**1**answer

131 views

### Equations and random subgroups in compact groups

EDIT: Here is a more specific question.
Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...

**4**

votes

**0**answers

120 views

### Is every compact monothetic group metrizable?

If $G$ is a compact (Hausdorff) topological group with a dense cyclic subgroup, is it necessarily true that $G$ is first countable? This claim seems to be implicit in a paper that I am reading at the ...