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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0answers
66 views

Does the compact-open topology retain topological groups?

Let $X$ be a topological space and $Y$ a topological group. Then $C(X,Y)$ is a group, and can also be endowed with the compact-open topology. Is $C(X,Y)$ in the compact-open topology necessarily a ...
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1answer
119 views

Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?

Let $G$ be a compact Lie group. An Abelian Lie subgroup $A \leq G$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $A'$ such that $A \leq A' \leq G$, then $A' = A$. Of course any ...
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Continuous surjection from $X(D_n)$ onto $\operatorname{Homeo}_0(D_n)$

Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...
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1answer
103 views

Reference request: placing a set with respect to the integer grid

For $x=(x_1,...,x_n)\in \mathbb{R}^n$, let $Q_x=(x_1,x_1+1)\times ...\times (x_n,x_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft here) the following ...
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1answer
105 views

Can you give me an example of a totally disconnected subgroup of a topological group that is not a topological group? [closed]

In the book The topology of fiber bundles, Steenrod characterize bundle over a base $X$ and totally disconnected structural group $G$ as follows. Theorem: Let $X$ be arcwise connected, arcwise ...
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2answers
84 views

Locally compact Polish groups acting on standard Lebesgue spaces

If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...
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2answers
170 views

Seeking to understand meaning of “von Neumann spectrum” in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page) I find that towards the end of the ...
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1answer
159 views

Residually finite group with dense finite index subgroups

Let $G$ be a locally compact Hausdorff topological group whose underlying abstract group is residually finite. Let $H\subset G$ denote the intersection of all finite-index, closed subgroups. Is there ...
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Is there an orbit map without path lifting property?

I am looking for an example of a topological group $G$ acting by homeomorphisms on a metrizable space $X$ such that the orbit map $X\to X/G$ doesn't have the path lifting property, that is, there is a ...
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Quantum analogue of certain property of compact groups

Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$. What is a precise description of a maximal ,or in some sense ...
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Criteria for density of subgroup of diffeomorphism group

Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
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206 views

Rational homotopy theory [closed]

I am trying to read the paper "Rational homotopy theory " by Quillen and am stuck with the notion of complete augmented algebra. He had defined the complete augmented algebra and I don't understand ...
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50 views

A Jacobian of the Cayley graph

Let $\pi$ be a finitely generated discrete group. Let $F$ be a finitely generated free group with an epimorphism to $\pi$. Let $K$ be the kernel of the epimorphism. Consider the Abelianization of $K$. ...
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Does the self-homeomorphism group of a finite CW complex have CW homotopy type?

Let $X$ be a finite CW complex and form the group $\mathcal{H}(X)$ of self-homeomorphisms $X\xrightarrow{\cong}X$, furnishing it with the compact-open topology. Under the assumptions on our space $\...
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1answer
169 views

Does a cocompact subgroup of a topological group contain a cocompact normal subgroup?

Motivation: It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$. Our question: Let $G$ be a topological ...
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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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1answer
593 views

For which G is BLG weak homotopy equivalent to LBG?

Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?)...
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1answer
592 views

Is there a compact, connected, totally path-disconnected topological group?

There exist homogeneous spaces such as the pseudo-arc, which are compact, connected, and totally path-disconnected. Is there a nontrivial, Hausdorff topological group with the same properties, i.e. ...
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89 views

Definition of reducible lattice

I am reading Raghunathan's book on discrete subgroups of Lie groups. In particular I am stuck on Corollary 5.19 which gives several equivalent conditions for a lattice in a semisimple Lie group to be ...
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1answer
115 views

Is each preseparable topological group narrow?

A topological group $G$ is defined to be $\bullet$ precompact if for any neighborhood $U\subseteq G$ of the unit there exists a finite subset $F\subseteq G$ such that $G=UF$; $\bullet$ narrow if for ...
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49 views

Decomposition into distal and proximal

For a topological group $G$ and a bounded real- or complex-valued function $f$ on $G$, the orbit closure of $f$ is the pointwise closure in the space of all bounded functions on $G$ of the orbit of $f$...
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1answer
258 views

How flexible is the infinite-dimensional torus?

Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group. Problem 1. Is it true that for ...
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1answer
90 views

Are the separability and autoseparability equivalent for (locally) compact topological group?

Definition. A topological group $G$ is called autoseparable if there exists a countable subset $S\subset G$ and a sequence $(f_n)_{n\in\omega}$ of automorphisms of $G$ such that for any neighborhood $...
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1answer
222 views

Different locally compact metrizable second countable topologies on the same group

Let $G$ be a non-Abelian infinite group. Can $G$ admit more than one (inequivalent) non-compact locally compact metrizable second countable topologies that make it a topological group? Thank you.
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How large this subset is to say that it should equal the group?

Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set $$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
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2answers
157 views

Empty interior of union of cosets?

The following question arises from trying to understand Lemma 1.3(ii) of arXiv:math/0405063. I believe a particular case of the proof (and in fact I think the proof is essentially equivalent to this ...
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46 views

Homomorphism defined on a dense subset extending in different way with respect to different target topologies

Let $G$ be a profinite group with a dense subset $S\subset G$. Let $H_1$ and $H_2$ be two topological groups whose underlying abstract group is the same. Can there exist homomorphisms of topological ...
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80 views

Nilpotency of topological groups

A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups $$ \{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G $$ ...
10
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1answer
657 views

Why are homeomorphism groups important?

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
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103 views

How a profinite group can be obtained from its normal open subgroups?

Let $\Delta$ be a set, each element of which is a profinite group (2 distinct elements of $\Delta$ may be isomorphic). Under what conditions on $\Delta$, there exists a profinite group $G$ which has $\...
8
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1answer
270 views

Equivariant cohomology of $\text{Diff}S^1/ S^1$ and Virasoro

Consider $$\mathcal{M}\ =\ \text{Diff}S^1/S^1$$ which is a contractible complex manifold with an action of $\text{Diff}S^1$ by translations. It is claimed in page 358 of [1] that $\mathcal{M}$ has ...
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79 views

Does the group of homeomorphisms of the hilbert cube have automatic continuity

A topological group is said to have automatic continuity if every homomorphism from it to a second countable topological group is continuous. Various topological groups are known to have this ...
10
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1answer
124 views

Is there a topological group with the small index property that does not have automatic continuity?

Here are the exact definitions of the terms: Let $G$ be a topological group. Then $G$ has the small index property if every subgroup of countable (including finite) index is open in $G$. Furthermore,...
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1answer
117 views

Topological analogue of an FC group?

By definition, a group is FC if all its conjugacy classes are finite. Has anything been published about a generalization of the FC property for topological groups?
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50 views

On self-duality of non-Archimedean local fields

The question to follow has already been asked by the OP at https://math.stackexchange.com/questions/3454735/on-self-duality-of-non-archimedean-local-fields. Due to a lack of feedback, the OP felt ...
7
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1answer
145 views

Permutation groups generated by finitely many point stabilisers

Assume that $G\leq\operatorname{Sym}(X)$ is a permutation group generated by all its point stabilisers, i.e. $G=\langle G_x \mid x\in X\rangle$. There is no cardinality restriction on $X$. Furthermore,...
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112 views

Are geometric progressions closed in the $p$-adic topology?

For a prime number $p$, the $p$-adic topology on the set $\omega$ of non-negative integers is generated by the base consisting of the arithmetic progressions $x+p^n\omega:=\{x+p^ny:y\in\omega\}$ where ...
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72 views

Homomorphism of composition to additive structure

Consider the following topological groups $\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
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1answer
177 views

Continuous semigroup homomorphism of composition to additive structure

Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
3
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1answer
186 views

Continuous function defined by measurable sets

Is the following slightly generalization of Corollary 20.17 in Hewitt and Ross Book (page 296) correct? Let $A$ be a subset of a profinite group $G$ ( compact, Hausdorff, totally disconnected ...
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2answers
298 views

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 $...
2
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0answers
83 views

Homomorphisms from circle to $GL(k,\mathbb{R})$ [duplicate]

Example 3 at the website tricki proves that every measurable homomorphism of groups from the circle to the non-zero complex numbers is continuous. Is there any analogous (true) statement for ...
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0answers
87 views

Continuous vs $L^2$ homomorphisms from circle to non-zero complex numbers

Let $T:S^1\to C^\ast$ be a group theoretic homomorphism from the circle to the non-zero complex numbers. Presumably it is true that if $T$ is $L^2$, then it is continuous. Is there a simple proof, or ...
7
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1answer
452 views

Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$

Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...
2
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0answers
143 views

Factoring a topological universal cover

Let $X$ be a compact, connected, locally path-connected, and semilocally simply connected topological group with $\pi_1(X) \cong \mathbb{Z}$. Let $u\colon \widetilde{X}\to X$ be its topological ...
2
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0answers
121 views

Regular epi- and mono-morphisms for locally compact (Hausdorff) groups

I am interested in what the regular monomorphisms are in the category of locally compact (for me, always Hausdorff) groups (with continuous group homomorphisms). It is easy to see that the equaliser (...
7
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1answer
174 views

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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1answer
123 views

$G$- space is locally compact [closed]

Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
3
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1answer
82 views

Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?

This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
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0answers
67 views

Irreducible unitary representations of discrete abelian groups

It seems to me that the statement below should be true but I would like to double-check. Statement: Let $H$ be a (separable) complex Hilbert space and consider its associated unitary group $U(H)$ ...

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