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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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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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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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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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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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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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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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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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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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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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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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
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...
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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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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 $\...
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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 ...
4
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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 ...
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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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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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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 ...
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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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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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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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1
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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
votes
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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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2
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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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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 ...
1
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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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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{\...
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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 ...
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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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1
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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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2
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Mackey theory in the setting of locally profinite groups
$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$.
Recall the following standard result [1, Thm. ...
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$G$- 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?
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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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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 $U(H)$ ...
5
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Compactly generated vertex stabilisers in compactly generated t.d.l.c. groups acting on trees
In the article cited below, I. Castellano gives a proof for the following result (Proposition 4.1).
Let $G$ be a compactly generated totally disconnected locally compact group. Suppose that $G$ ...
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Question about regular representation of compact group
I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
1
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0
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Bruhat cell of a Coxeter element
If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a ...
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1
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Relation to the Bruhat cell
Let $g\in\operatorname{SL}_n(\mathbb Z)$ such there exists $v\in\mathbb Q^n$
such that $v, gv, \dotsc, g^{n−1}v$ is a $\mathbb Q$-base of $\mathbb Q^n$ and there exists a $\mathbb Z$-base $w_1, \dotsc,...
2
votes
1
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Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
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Are $T_0$ topological quasigroups completely regular?
In 1957 H. Salzmann generalized to quasigroups but weakened the standard result that $T_0$ topological groups are completely regular. He was able to show that $T_0$ topological quasigroups are regular ...
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The rigidity of the countable product of free groups
For a natural number $n$ let $F_n$ be the free group with $n$ generators.
The group $F_n$ is endowed with the discrete topology.
Given an increasing sequence $\vec p=(p_k)_{k\in\omega}$ of prime ...
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Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
2
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How to prove that Chevalley groups over $\mathbb R$ have no compact factors
I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem.
I've been told in another thread ...
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Describing compact Lie groups in purely topological terms
Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, ...
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Polish groups with no small subgroups
Definitions.
A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space.
A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity ...
3
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1
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Reduction of the structure group of $\mathbb{R}^n$-fiber bundles to a special subgroup of $\mathrm{Homeo}(\mathbb{R}^n)$
Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$
In other words, $G$ is the group of all equivariant self-...
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1
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When are function groups monothetic?
Let us consider the group of continuous functions $C(S^1, S^1)$ from the circle to itself with the compact open topology. Does it have a chance to contain a dense cyclic subgroup?
Unfortunately it ...
6
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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 ...
2
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Factorization of characters
Linked to the end of this question here and because the subject involves many deformations of shuffle, I came to the following
Let $k$ be a $\mathbb{Q}$-algebra and $\mathfrak{g}$ a $k$-Lie algebra, ...