The topological-groups tag has no usage guidance.

**22**

votes

**0**answers

939 views

### Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...

**14**

votes

**0**answers

950 views

### What groups are Lie groups?

We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...

**14**

votes

**0**answers

502 views

### Closed connected additive subgroups of the Hilbert space

It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...

**10**

votes

**0**answers

202 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 ...

**8**

votes

**0**answers

129 views

### A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...

**8**

votes

**0**answers

263 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 ...

**8**

votes

**0**answers

417 views

### Are there non-reflexive abelian topological groups isomorphic to their second dual?

I posted the following question in a comment at
Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...

**7**

votes

**0**answers

243 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 ...

**6**

votes

**0**answers

247 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 ...

**6**

votes

**0**answers

399 views

### Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...

**5**

votes

**0**answers

111 views

### Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...

**5**

votes

**0**answers

75 views

### Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$.
...

**5**

votes

**0**answers

216 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 ...

**5**

votes

**0**answers

174 views

### Shrinking Group Actions

This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a ...

**4**

votes

**0**answers

135 views

### Examples of a topological semidirect product

Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes ...

**4**

votes

**0**answers

69 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 ...

**4**

votes

**0**answers

109 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 ...

**4**

votes

**0**answers

360 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 ...

**4**

votes

**0**answers

91 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 ...

**4**

votes

**0**answers

2k views

### Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups

Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?

**3**

votes

**0**answers

179 views

### Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...

**3**

votes

**0**answers

132 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

**0**answers

329 views

### Topological proof of a result in Logic

I proved the result below using logic. My questions:
Can this theorem be proved by purely topological means?
Do you know any theorems that either can be used to prove the same result, or which give ...

**3**

votes

**0**answers

127 views

### Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?

**3**

votes

**0**answers

210 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 ...

**2**

votes

**0**answers

98 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 $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. ...

**2**

votes

**0**answers

171 views

### Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
...

**2**

votes

**0**answers

94 views

### Free profinite completions

Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?

**2**

votes

**0**answers

425 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 $G$ ...

**2**

votes

**0**answers

125 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 ...

**2**

votes

**0**answers

132 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 ...

**2**

votes

**0**answers

172 views

### Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...

**1**

vote

**0**answers

22 views

### In a topological group $G$ with its lower uniformity, if $G$ is locally totally bounded, is its completion locally compact?

There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ...

**1**

vote

**0**answers

96 views

### Hausdorff topologies on Q

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

**1**

vote

**0**answers

28 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 ...

**1**

vote

**0**answers

42 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

**0**answers

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 ...

**1**

vote

**0**answers

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

**0**answers

161 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 ...

**1**

vote

**0**answers

456 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 ...

**1**

vote

**0**answers

198 views

### exotic compact group

Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...

**0**

votes

**0**answers

40 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 ...

**0**

votes

**0**answers

21 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 ...

**0**

votes

**0**answers

49 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 ...

**0**

votes

**0**answers

45 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 ...

**0**

votes

**0**answers

79 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 ...

**0**

votes

**0**answers

26 views

### topologically finitely generated residually finite group

Suppose that $G$ is a topologically finitely generated profinite group and $H$ is a subgroup of countably infinite index. Can I say that $H$ must be topologically finitely generated with the subspace ...

**0**

votes

**0**answers

46 views

### connected Polish groups

We know that a connected locally compact Hausdorff topological group is a pro-Lie group, by the Gleason-Yamabe theorem. Is there a known characterisation of the connected Polish groups?

**0**

votes

**0**answers

48 views

### The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...

**0**

votes

**0**answers

114 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) ...