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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Groups with no (proper) closed subgroups?
$\mathbf{Z}$ with the profinite topology has the property that every subgroup is closed (Topological groups in which all subgroups are closed). What topological groups have the property that no (...
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With a linear representation, how does the continuity of $G \to \mathrm{GL}(V)$ relate to that of $G \times V \to V$?
I'm currently reading Traces of Hecke Operators by Knightly and Li, while simultaneously revisiting the adelic/representation-theoretic point of view on automorphic forms.
In Knightly and Li, they ...
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Model structure on the category of topological groups
Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...
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Bohr compactification as a topological compactification
Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$.
Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is ...
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Naive question on adelic groups
The ever-reliable Wikipedia says:
... an adelic algebraic group is a semitopological group defined by...
No more details are given, and I was wondering if the multiplication only being separately ...
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Inverses in convolution algebras
Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...
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About Lie group $G$ has this escape property?
Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
...
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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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Topologies that turn the real numbers into a compact Hausdorff topological group
If I'm not mistaken, the question I put on the title used to be on this site, but I'm not being able to find it at all. I'm therefore reposting it so that someone can either give me the old link or ...
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Intersection of all open subgroups vs. the intersection of all open normal subgroups
I am interested to know examples of topological groups $G$ for which the intersection $\bigcap\{H\leq G\mid H\text{ open}\}$ of all open subgroups of $G$ is the trivial subgroup but for which the ...
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Discrete spectrum and almost periodicity
According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to \...
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Topology on cohomology of a sheaf of topological groups
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
Is there a natural way to introduce topology on $H^i(X,...
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$\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 ...
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Examples of totally disconnected, locally compact non-sigma-compact groups
I am looking for examples of totally disconnected, locally compact groups, which are not sigma-compact. For a start any such an example would do, so that I can a feeling for those groups and how to ...
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Solid tensor product of pro-discrete space with Laurent series
Consider the category $\operatorname{Solid}_{\mathbf{Z}}$ of solid abelian groups in the sense of Clausen-Scholze. This category is a full subcategory of condensed abelian groups, $\operatorname{Cond}...
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A property of rapid sequences of natural numbers
$\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
Taras Banakh and me proceed a long quest answering a question of ougao at ...
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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 ...
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Topological groups defined by completely disconnected subgroups
Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
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Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$
Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:
$$
\| T_t : L_p(\mu) \to L_q(\mu)\| \...
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Semigroup product of the left-invariant completion of a Polish group (restatement of Question 71389)
This is a re-statement, of sorts, of the question Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?, so far unanswered.
...
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Uniqueness of left-invariant Borel probability measure on compact groups
On a compact topological group, consider two left-invariant probability measures $\mu$ and $\nu$ defined on the Borel sigma-algebra. Is it true that they coincide?
It is classical that the Haar ...
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Irreducible representations of compact groups
Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ${\...
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Homomorphisms of Topological Groups which are Automatically Fiber Bundles?
Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...
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Hausdorff distance in compact Lie groups
Let $G$ be a compact Lie group with a compatible biinvariant metric $d$. The hyperspace $K(G)$ of nonempty compact subsets of $G$ is a compact metric space with the Hausdorff metric, and it is easy to ...
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Pontriagin reflexivity of the character group
For an Abelian topological group $G$ by $G^{\wedge}$ we denote the Pontryagin dual of $G$, i.e. the group of continuous homomorphisms $G\to\mathbb T:=\{z\in\mathbb C:|z|=1\}$. The group $G^{\wedge}$ ...
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Extending the topology on a set to the group/vector space it generates
The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form
$2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$
The tuple ...
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Can each non-open analytic subgroup of a Polish abelian group be covered by countably many closed Haar null subsets?
By a result of Laczkovich ('Analytic subgroups of the reals' Proc AMS Vol 126 (1998)), any non-open analytic subgroup of a Polish locally compact group can be covered by countably many closed Haar ...
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Steinhaus number of a group
$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$.
Let $\mathcal A_X$ be the family of ...
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Growth function of locally compact groups
Every locally compact second countable group $G$ has a regular left-invariant measure $h$, the Haar measure. On the other hand the Birkhoff–Kakutani Theorem asserts that such groups also admit a ...
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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 ...
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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 ...
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Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?
Let $G$ be a locally compact (second countable) group and let
$$
G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the ...
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How big $|\operatorname{Aut}(M)|$ can be, given $|\partial\operatorname{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 $\...
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Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
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What can lattices tell us about lattices?
A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
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Countable companions for Polish locally compact groups and their orbit equivalence relations
In "Countable sections for locally compact group actions" (Ergod. Th. & Dynam. Sys., 1992), Kechris proved that if $G$ is a Polish locally compact group acting in a Borel way on a ...
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When exponential map is 1-1 from vector fields to diffeomorphisms
Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
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Does every compact abelian group contain a Kronecker set generating a dense subgroup?
Let $G$ be a compact metrizable abelian group with infinite exponent.
Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
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How to make an endomorphism of an LCA group invertible
Consider a pair $(G,\phi)$ where $G$ is a (discrete) Abelian group and $\phi\colon G\to G$ is an endomorphism of $G$. There is a usual trick to construct a new pair $(G',\phi')$ with the property that ...
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Special monomorphism to encode the inclusion of topological submonoids
Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms.
Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with ...
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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 ...
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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 ...
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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 ...
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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?
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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 there ...
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On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
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Is each locally compact group topology on the permutation group discrete?
Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete?
Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
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Is any continuous group homomorphism from R to C* an exponential map? [closed]
Consider $\mathbb{R}$ to be an additive topological group, and $\mathbb{C}^{\ast}$ to be a multiplicative topological group.
Is the following statement true?
If so, then how can one prove it?
...
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Does the group G(K) have a cocompact solvable closed subgroup?
Let $K$ be a (locally compact) local field and $G$ be a linear algebraic $K$-group.
Does the topological group $G(K)$ have a cocompact solvable closed subgroup?
If $\mathrm{char}(K)=0$, it is true ...
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Sylow subgroups of abelian profinite groups
If $G$ is a finite abelian group, then we have a decomposition
$$G\cong \prod_{p} G(p)$$
where $G(p)$ is the $p$-Sylow subgroup of $G$. This product makes sense as for all but finitely many primes $p$,...