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).
257 questions with no upvoted or accepted answers
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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 ...
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732
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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 ...
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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
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The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?
For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
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Can Rep(G) tell us whether G is discrete?
Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...
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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 ...
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Is the group $\operatorname{Diff}^1_0(\mathbb R^d)$ connected?
Is the group
$$ \operatorname{Diff}^1_0(\mathbb R^d) = \operatorname{Diff}^1(\mathbb R^d) \cap \big(\operatorname{Id}_{\mathbb R^d} + C^1_0(\mathbb R^d,\mathbb R^d)\big) $$
connected? Here
$$ C^1_0(\...
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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 ...
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Metric completion of an algebraically closed field is algebraically closed?
Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed?
We can ...
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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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Does each compact topological group admit a discontinuous homomorphism to a Polish group?
A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...
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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^\...
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Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?
Let $G$ be a finite group. It has been shown that:
If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
If the probability ...
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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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"Homotopy homomorphisms" of homeomorphisms of Euclidean space
For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...
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What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
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435
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The group analogue of the James construction
If $(X,e)$ is a based topological space, then the James construction on $(X,e)$ is the free topological monoid with unit $e$: $J(X)=\coprod_{n\geq 1}X^n /\sim $ where $(x_1,...x_{j-1},e,x_j,...,x_n)\...
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Is every space a classifying space?
Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological ...
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'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 ...
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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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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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Second homotopy group of a topological group
It is well-known that any Lie group $G$ has $\pi_2(G)=0$: see this question. Is the same true for any compact (Hausdorff) topological group? Or even for locally compact ones? Maybe there is a way of ...
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306
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Has the Roelcke completion of a topological group any reasonable algebraic structure?
It is well-known that each topological group $G$ carries (at least) four natural uniformities:
the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...
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example of an n-transitive but not infinitely transitive group action on a space
Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ ...
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Is there an countable amenable dense subgroup of $U(\ell^2 \mathbb N)$?
Question: Does the unitary group $U(\ell^2 \mathbb N)$, equipped with the strong operator topology, contain a countable dense subgroup which is amenable as a discrete group?
I would be also ...
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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 $\...
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An abelian category with a full embedding from topological abelian groups
I know this is a very vague question, but I can't think of a better question to ask.
Consider the category $\mathscr{C}$ of pro-sets created by diagrams containing only injections. The objects $A: \...
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Degree of automorphic forms, SL(3,Z), and the elliptic Gamma function
In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as
$$
\Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-...
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Is each Choquet topological group strong Choquet?
A topological space $X$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game.
It is known that a metrizable space $X$ is
$\bullet$ Choquet if and only if it ...
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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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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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Is each completely minimal topological group minimal?
A topological group $G$ is called
$\bullet$ minimal if it admits no strictly weaker Hausdorff group topology;
$\bullet$ completely minimal if it is Raikov-complete in each weaker Hausdorff group ...
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Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$
What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?
I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
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315
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Two different Thom diagonals in recent literature?
Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...
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419
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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 $...
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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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Dependence on Urysohn's Lemma in Cartan's Construction of Haar Measure
This question was posted by someone else on stackexchange three months ago, but no one has answered as of yet:
Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence ...
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About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
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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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341
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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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47
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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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365
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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 ...
- 1,396
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293
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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 ...
- 4,728
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2k
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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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192
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When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?
The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces.
I am looking for classes of examples of connected topological groups/...
- 1,174
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132
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Is $\mathbf{C}_p(X)$ self-dual?
Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
- 1,485
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376
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Topologies on diffeomorphisms groups
Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual ...
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