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).
756 questions
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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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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 ...
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Eigenfunction of ergodic skew product fixed by commutator?
Background: Let $(Y, \mathcal{B},\mu,T)$ be an ergodic probability system and let $G$ be a compact metrizable group with compact subgroup $H$. Given a measurable map $\rho:Y \to G$. We may define the ...
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locally topologically finitely generated t.d.l.c. group
I am trying to find an example of the following situation.
$G$ is a t.d.l.c. (totally disconnected locally compact) $\sigma$-compact topological group in which every compact open subgroup is ...
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Are convolution algebras ever "topologically noetherian"?
For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means ...
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Topologically finitely generated residually finite group
Suppose that $G$ is a topologically finitely generated profinite group,
and let $H$ be a subgroup of countably infinite index. Is $H$ necessarily topologically finitely generated with the subspace ...
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countably-infinite-index subgroup of a finitely generated profinite group
Suppose that $G$ is a profinite group with the property that every open compact subgroup is topologically finitely generated and just infinite. Suppose that $H$ is a commensurated subgroup of $G$ with ...
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If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer ...
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If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
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discrete group cohomology vs continuous group cohomology for profinite groups
Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map $...
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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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Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods
Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
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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 $\mathcal A$ ...
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algebraic groups over non-archimedean local fields acting on buildings
I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and ...
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query about quasi-simple algebraic groups over local fields
Suppose that $G$ is an absolutely quasi-simple algebraic group over a non-archimedean local field $k$ (of either zero or positive characteristic). Is it known whether or not it is necessarily the case ...
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Distributivity of group topologies on $\Bbb Z$
Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
[1] ...
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Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$
Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function
$$f:G\times G\to G$$
$$f(x,y)=xy^{-1}$$
is continuous at $(1,1)$?
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Are infinite groups "locally topologizable"?
Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
The question is inspired by and related to this ...
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A group allowing exactly 7 group topologies
Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$...
- 1,145
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Parallel group topologies on Prüfer groups
Let $p$ be a prome number. Are there group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that
$$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$
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Equations and random subgroups in compact groups
EDIT: Here is a more specific question.
Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...
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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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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$.
...
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Infimum of two group topologies
Let $G$ be a non-abelian group and $\mathcal S$ and $\mathcal T$ be group topologies on $G$. What is the largest group topology $\tau$ on $G$ with $\tau \subseteq \mathcal T\cap \mathcal S$?
In ...
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Is an extension of compact Hausdorff topological groups compact?
Let $1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1$ be a short exact sequence of topological groups (i.e., all maps are continuous, $A = \mathrm{Ker}(c)$, and $C = \mathrm{Coker}(a)...
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Union of conjugates of a closed subgroup of a compact group
Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$.
Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of $G$...
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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 ...
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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
$$...
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Simple but topologizable [closed]
Do you have an example of an infinite simple group with at least 3 distinct group topologies on it?
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Locally compact vs. compactly generated in group theory
I'm studying the Creutz-Shalom paper on the normal subgroup theorem for dense commensurators of lattices. The authors state their results for locally compact and compactly generated groups. I do not ...
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Existence of infinite groups that are too reluctant to be topological
With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
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The group of diffeomorphisms with compact support
Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My ...
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In what topological abelian groups does convergence to zero imply summability?
(This question has been on math.SE for over a week and has not gotten any answers.)
Let $G$ be a (T$_0$) topological abelian group, and let $0$ be its identity element.
Assume that for all index ...
user5810
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Compact Quantum Groups and the Existence of the Classical Haar Measure
Before I state my question, let me provide the definition of a compact quantum group.
Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if
$ ...
user36116
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Haar measurable sets and quotient maps
Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
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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?
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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 $n$?
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Action of a profinite group
Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
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Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)
I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...
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When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube
Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus ...
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rationalization of classifying spaces
This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an $H$-...
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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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Automorphism group of compact abelian group
I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as ...
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Hausdorffness inheritance in topological groups
Suppose $\mathcal T$ and $\mathcal S$ are two compatible Hausdorff topologies on a group $G$ and $\mathcal R$ is a maximum compatible topology on $G$ with $\mathcal R \subseteq \mathcal T\cap \mathcal ...
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Connectedness properties of groups of homeomorphisms
Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...
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Group of homomorphisms with real target and circle target
Denote by $\mathrm{Hom}$ continuous group homomorphisms. Fix the quotient homomorphism $\mathbb{R}\to S^1$.
Can one characterize those topological (e.g., locally compact, and in particular discrete) ...
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injective implies completion injective?
Background and definitions.
Let $k$ denote a field complete with respect to a non-trivial non-archimedean norm. Let $R$ be the integers in $k$, and say $\pi\in R$ with $0<|\pi|<1$ ($\pi$ doesn'...
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Locally finite compact groups
I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...
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Is a weakly separable group always Lindelöf?
By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
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Restriction of "$\pi_{1}$" to topological groups
Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$?
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