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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Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
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Restrictions on pointed lifts of isometries
Let $M$ be a (closed) Riemannian manifold and let $f$ be an isometry of $M$ fixing a point $\ast \in M$ that acts trivially on $\Gamma := \pi_1(M,\ast)$.
Then there is a unique isometry $\tilde{f}$ of ...
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Finite covolume of uniform lattice in quotient group
Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact.
Suppose ...
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fibre sequence of classifying space
I read Steve Mitchell's Notes on principal bundles and classifying spaces (pdf).
There is a theorem: Let $G$ be any topological group, $H$ an admissible
normal subgroup. Then there is a homotopy-fibre ...
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Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig
"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
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Haar measure coming from Pontryagin duality v/s Fourier inversion
Not research but advertising this question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
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Morphism of R-algebras between R-adic algebras
Let $R$ be a $f$-adic ring. Let $A$ and $B$ be $R$-adic algebras. I would like to show that any morphism of $R$-algebras between $A$ and $B$ is actually adic.
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Definition of a continuous Gabor frame
I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\...
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Left-side cosets of an open subgroup
Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}\in G$. We assume $L\cap g_{1}K\neq \emptyset$ and $L\cap g_{2}...
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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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Do Locally Contractible, Path-Connected Groups have Accessible Bases?
Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\...
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Countable sum $\bigoplus_{n=0}^\infty\mathbb Z_p$ as a topological group
$\DeclareMathOperator\colim{colim}$This is inspired by Clausen's answer.
Question: Recall that $\mathbb Z_p$ is endowed with the $p$-adic topology. Consider the countable sum $M:=\bigoplus_{n=0}^\...
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Question about additive subgroups of the real line and the density topology
I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question.
Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $E\...
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Haar mesure on $\mathrm{GL}_{d}(F)$
$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...
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On measurability of certain group actions on spaces of bounded measurable functions
Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded ...
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CH and the density topology on $\mathbb{R}$
In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
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Is there a Hausdorff space that is also a group such that the group operation is continuous but the inversion map is not continuous?
The question is from the definition of to topological group. I can find an example such that the inversion map is continuous but the group operation is not continuous, but I cannot find an example ...
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Group completion of topological monoids
Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\...
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Pointed versus unpointed maps into a topological monoid
I've just stumbled on something that seems either too good to be true,
or else too good for me not to have heard of it before.
It has to do with the basepoint forgetting map
$$
u: [A, M] \to \langle A,...
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About locally compact groups without compact subgroups
Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
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Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean?
Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I ...
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A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$
Let we have a complex of abelian topological or lie groups $$\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 ...
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Characteristically simple locally compact abelian groups
Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `...
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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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Are there locally compact groups which have no compact open subgroups and no discrete infinite cyclic subgroups?
The posting of this question was suggested by Yemon Choi: see Discrete cyclic subgroup.. The question is not mine; it's just a rephrasing of Discrete cyclic subgroup.
EDIT 4. This post claims that ...
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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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Non-measurable sets on groups from translation invariance
The most well-known construction of a non-measurable set is the Vitali set. The idea behind Vitali sets is to split up the space (such as $[0,1]$) into equal-sized copies (guaranteed by translation ...
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Splendid groups
The following definition has arisen naturally in two papers of mine. The papers are on rather unrelated topics; of course they are within my narrow interests, so there's some symbolic dynamics ...
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The Tychonov cube $X^X$ of a compact space $X$ is a compact semigroup with the composition operation
Reading a book about Ramsey theory this is the first example of a compact (semitopological) semigroup, which is a nonempty semigroup S with compact Hausdorff topology for which $x \mapsto x*s$ is a
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Making use of extra symmetries; more examples?
TL; DR.
In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I ...
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How to prove continuity in topological group action of ${\rm{GA}}_a(X)$ on $T(X)$, to make ${\rm{GA}}(X)$ a topological group?
The question comes from the following paragraph of a text on geometry in the context of affine geometry (Marcel Berger et al., "Geometry I", P56-57):
2.7.1.3. If we don't want to resort to ...
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Positive type function on open subgroup
Let $\phi: G \rightarrow \mathbb{C}$ be a continuous function. We say that $\phi$ is positive type if $\sum_{i,j=1}^{n} c_i\bar{c_j}\phi(g_{j}^{-1}g_i)\geq 0$ for all $n \in N, c_i \in \mathbb{C}, g_i ...
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What is the smallest number of nowhere dense affine subsets covering a topological group?
$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$.
Given a non-discrete topological ...
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Algebraic rigidity in the automorphism group of the Cantor set
Let C be a Cantor set (middle third). Now we know that C is a totally disconnected compact topological space with the natural topology (i.e., $C=\{0,1\}^{\mathbb{N}}$). Let G:=Homeo(C) be the set of ...
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Two cardinal characteristics of the continuum, related to the Bohr topology on integers
For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in ...
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Topological groups in which all subgroups are closed
General question: does there exist a nondiscrete topological group $G$ such that all subgroups of $G$ are closed? Or, does there exist a nondiscrete topological vector space $V$ such that all vector ...
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Complete topological groups in which all subgroups are closed
My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation.
General question: does ...
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Mapping property of $p$-Sylow groups of profinite groups
Let $G$ be an abelian profinite groups. Then we have the Sylow group decomposition
$$G\cong \prod_p G_p.$$
In the case of finite groups, we have $ \prod_p G_p\cong \bigoplus_p G_p$ and thus
$$\text{...
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Unitary representation is strictly continuous
Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology ...
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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 ...
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kernel and cokernel of corestriction map in cohomology of a profinite group
Let $G$ be a profinite group, $N$ a normal open subgroup and $A$ a discrete $G$-module. We have a corestriction map $cor: H^1(N, A)_{G/N} \to H^1(G, A)$. Are there any results on the kernel and ...
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Is there a natural topology on $\mathbb{C}(t)[x_1,\ldots, x_n]$ with this property?
Is there a good topology on $A=\mathbb{C}(t)[x_1,\ldots, x_n]$ so that $A$ is a topological algebra with the following property:
For any $N>0$ and a polynomial $F\in\mathbb{C}[x_1,\ldots, x_n]$ ...
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Integration in a finite dimensional vector space
Let $V$ be a finite dimensional complex vector space. Let $G$ be a compact group with normalized Haar measure $\mu$. In the representation theory of compact groups, I encounter
$$\int_G f(g) \mu(dg)$$
...
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Sufficent condition for strict morphism of normed vector spaces
Let $K$ be a non-archimedean field of char 0 and a morphism $f:V \rightarrow W$ of normed $K$-vector spaces given. The map $f$ is said to be strict if $V/\ker(f)$ with the quotient topology is ...
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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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Group of surface homeomorphisms is locally path-connected
I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance ...
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Is $X$ closed in $Aut_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])$?
Consider $\mathbb{C}$-algebras
$$A = \mathbb{C}[t][x_1,\ldots, x_n]\subset\mathbb{C}(t)[x_1,\ldots, x_n] = B$$
Group $\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])$ carry a power series topology (...
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Is the commensurator of a tree lattice a simple group?
Let $T$ be an $n$-regular tree ($n\geq3$). Let $\operatorname{Aut}^+(T)$ be the subgroup of index 2 of $\operatorname{Aut}(T)$ preserving the bicoloring of the tree for which adjacent vertices have ...
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Countability of conjugacy classes in profinite groups
In the MOF question [1] it was asked if $G$ is a second-countable profinite group with uncountably many subgroups, does it follow that it has uncountably many closed subgroups modulo conjugacy?
A ...
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Spaces of closed subgroups of a profinite group up to conjugacy
$\DeclareMathOperator{\Sub}{\operatorname{Sub}}$ Let $G$ be a profinite group and consider the space $\Sub(G)$ of closed subgroups of $G$ equipped with the profinite topology. That is, we have $G = \...
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