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Questions tagged [topological-groups]

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated

I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent: 1) Every open subgroup $\Gamma_0$ is topologically finitely generated. 2) The ...
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1answer
212 views

What are the almost periodic functions on the complex plane?

The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
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1answer
94 views

Separable topology on a group

In the paper "Continuous isomorphisms onto separable groups", Applied General Topology, (13) 2012, 135--150, L. Morales Lopez proved Theorem: Let $G$ be an Abelian group with $|G| \leq 2^{2^{\...
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1answer
70 views

continuity of certain map which is defined on a Stonean space

Let $G$ be a discrete group which acts continuously on a Stonean space $\Omega$. Consider the map $f\colon \Omega\to \{0,1\}^G$ sending $x\in \Omega$ to $\chi_{G_x}$, where $\chi_{G_x}$ denotes the ...
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1answer
310 views

Classification of compact connected abelian groups

It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
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Which countable discrete groups have a metrisable group compactification?

Let $G$ be a countable discrete group. A group compactification of $G$ is a compact Hausdorff topological group $H$ such that there is a group homomorphism $\iota\colon G\to H$ with dense image. For ...
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1answer
170 views

Every group of totally disconnected type is locally profinite?

Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods. Let's call this a group of totally disconnected (td)-type. On the other hand, we ...
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1answer
148 views

Finite dimensional compact abelian group that is not a product of connected and a totally disconnected

Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is ...
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1answer
172 views

Definition of Haar integral in Bushnell and Henniart

In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional $$ I: C^{\infty}_{c}(G) \...
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A generalization of the character group

Let $G$ be a group. We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$ where $Tor(\mathbb{T})$ is the group of torsion elements of the unit ...
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A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters

Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$? (...
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A ZFC-example of a countably compact paratopological group which is not a topological group

Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group? (The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...
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141 views

Appearances of $\mathbb{Q}/\mathbb{Z}$ in Pontryagin duality for profinite groups

(This is a somewhat lazy question which came up as I'm reading about Pontryagin duality for the first time) For a locally compact abelian topological group $G$, its Pontryagin dual is the group of ...
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1answer
77 views

Measure on group invariant under group action on metric space

This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO. The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...
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2answers
332 views

Ostrowski's Theorem for topological rings?

Ostrowski's theorem classifies all absolute values on a number field $K$. Questions: More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field? In ...
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1answer
113 views

The completeness of spaces of continuous functions with the compact-open topology

For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology. Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
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Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...
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Interior of fundamental domains of lattices in locally compact groups

Let $G$ be a locally compact abelian group, and let $\Lambda$ be a lattice in $G$, i.e. a discrete subgroup such that the quotient group $G/\Lambda$ is compact. A fundamental domain for $\Lambda$ in $...
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1answer
251 views

What does the $p$-adic closure of an arithmetic lattice look like?

Let $\Gamma$ be an arithmetic lattice in a linear algebraic $\mathbb{Q}$-group $\mathbf{G}$, that is, $\Gamma$ is a subgroup of $\mathbf{G}(\mathbb{Q})$ that is commensurable with $\mathbf{G}(\mathbb{...
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0answers
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Fixed-point properties for affine actions of topological groups

T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group $G$. Two of them are: (F2) Every jointly continuous ...
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Is the image of a locally compact totally disconnected group also locally compact and totally disconnected? [duplicate]

Let $G$ be a locally compact totally disconnected group and let $\phi$ be a surjective homomorphism from $G\to H$ (added later: where $H$ has the topology coinduced by $\phi$). Is H also locally ...
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Currently active Research Areas in Topological Groups

I have done my Master's degree in mathematics and currently I am doing Ph.D. in mathematics in India. I have completed my course work and my supervisor works in Topological Groups. He suggests me some ...
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1answer
183 views

homeomorphisms induced by composant rotations in the solenoid

Let $S$ be the dyadic solenoid. Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$. $X$ is called a composant of $S$. It is well-known ...
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Weighted cancellation norm of a word computation

A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$. We say that two disjoint pairs $\{...
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1answer
123 views

When does a map of spaces deloop a closed subgroup inclusion?

I believe Kan showed that any connected CW complex is the delooping of a topological group. I'm interested in the relative question: Question: Let $Y \to X$ be a map of connected CW complexes. Under ...
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1answer
191 views

If $G$ is a paracompact topological group, then is $G \times G$ paracompact?

If $G$ is a paracompact topological group, then is $G \times G$ paracompact? This question is raised by Gepner and Henriques (first paragraph of 2.2). Of course, this is not true for arbitrary ...
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0answers
143 views

A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
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2answers
88 views

A reasonable topology on the group of minimal usco maps

An usco map is an abreviation for an upper semicontinuous multi-valued map with non-empty compact values. An usco map $f:X\multimap \mathbb R$ is called minimal is it coincides with each usco map $g:X\...
3
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0answers
196 views

The closed subgroup of the idele corresponding to the maximal elementary $p$-extension of a global field

I want to know whether for any global number field $k$, the closed subgroup of the idele corresponding to the maximal elementary $p$-extension ($p$ is a prime number) is $k\ J^p$. The critical point ...
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1answer
115 views

Has the Erdős space the structure of a monothetic topological group?

This question is motivated by this MO-problem asking if the Erdős spaces $\mathfrak E$ and $\mathfrak E_c$ admit a self-homeomorphism with dense orbits of points. The affirmative answer would follow ...
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1answer
78 views

A question on quasitopological group

Suppose that $G$ is a regular feebly compact Moore quasitopological group. Must $G$ be a topological group? This was previously posted here on MathSE also. A semitopological group $G$ is a group $G$ ...
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1answer
172 views

What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?

For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian): finite groups $\leftrightarrow$ ...
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79 views

Fixed point set with non-empty interior

Let $G$ be an infinite compact separable Hausdorff metric group, and $H\subset G$ a closed subgroup, such that the left $G$-action on $G/H$ is effective (i.e., $H$ doesn't contain a non-trivial closed ...
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2answers
161 views

The number of distinct closed subgroups of a compact monothetic group

Let $G$ be a connected compact separable Hausdorff metric group, which is monothetic, i.e., has a dense subgroup generated by a single element. Such a group is necessarily Abelian. Question: Can the ...
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1answer
261 views

Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$

I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$. Here we can take either: $B^...
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3answers
143 views

Closed free subgroups of the automorphism group of the countable atomless boolean algebra

Let $\mathcal{B}$ be the (unique up to isomorphism) countable atomless boolean algebra, and $\mathrm{Aut}(\mathcal{B})$ its automorphism group, with pointwise convergence topology. My question: Does $...
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1answer
102 views

Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally compact Abelian groups?

It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$. Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or ...
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1answer
298 views

Are all compact subsets of Banach spaces small in a measure-theoretic sense?

Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
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1answer
159 views

Group completion of topological monoids

Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\...
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Sum-sets of sets of positive measure in the Hilbert cube

Problem. Let $\lambda$ be the standard product measure on the Hilbert cube $[-\frac12,\frac12]^\omega$ and $A,B$ be two $\lambda$-positive Borel subsets of $[-\frac12,\frac12]^\omega$. Is it true ...
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102 views

A section over an orbit space

Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup. Questions: ...
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1answer
128 views

Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$

One can define the convergence of a sequence $(\Lambda_k)_k$ of full rank lattices as folow : $(\Lambda_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff \forall k\in \mathbb{N} ,\exists ...
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Countable elementary sub-structure of the automorphism group of the binary rooted tree

Let $G$ be the automorphism group of the binary rooted tree. The downward Löwenheim-Skolem theorem states that G has a countable elementary sub-structure. My question is whether such sub-structure ...
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0answers
77 views

Maximal subgroups of infinite index and profinite completion

Preliminary remark: I'm mainly interested in an answer (or link to ressources) in the specific context of the first Grigorchuk group, but I believe that it may be of some interest to state the ...
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2answers
166 views

The disintegration of the convolution of two probability measures

Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
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1answer
54 views

Convergence of some object depending on functions with compact support

Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ ...
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1answer
58 views

Haar-$\mathcal{I}$ set and Polish groups

Let $\mathcal{I}$ be a semi-ideal of sets with empty interior on a compact metrizable space $K$. Let an $F_σ$-set $A$ in a Polish group $X$ generically Haar-$\mathcal{I}$. Then is $A$ always ...
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0answers
29 views

closed and bounded intervals of definably complete ordered groups

True or False? All closed and bounded intervals of definably complete ordered groups are definably compact. Let $G$ be an ordered abelian group. Then, a definable subset $D ⊆ G$ is said to be ...
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118 views

Hypercohomology of topological abelian groups

Suppose $X$ is a scheme, and $A, B$ two étale sheaves of locally compact topological abelian groups. Assume there is a map of étale sheaves such that for any $f: U\to X$ étale, $\Gamma(U_{\rm ét}, A)\...
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Do maximally almost periodic groups embed homeomorphically into their Bohr compactifications? [duplicate]

If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is ...