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).

Filter by
Sorted by
Tagged with
7 votes
0 answers
263 views

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-...
FractalScout's user avatar
8 votes
1 answer
675 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 $LBG \sim BLG$?)...
David Roberts's user avatar
  • 33.4k
13 votes
1 answer
762 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. ...
Jeremy Brazas's user avatar
0 votes
0 answers
248 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 ...
user551642's user avatar
3 votes
1 answer
139 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 ...
Taras Banakh's user avatar
  • 40.7k
1 vote
0 answers
72 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 $f$...
user95282's user avatar
  • 997
14 votes
1 answer
557 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 ...
Taras Banakh's user avatar
  • 40.7k
1 vote
1 answer
136 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 $...
Taras Banakh's user avatar
  • 40.7k
4 votes
1 answer
287 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.
Bedovlat's user avatar
  • 1,939
0 votes
0 answers
97 views

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}^...
Meisam Soleimani Malekan's user avatar
5 votes
2 answers
235 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 ...
Matthew Daws's user avatar
  • 18.5k
1 vote
0 answers
123 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 $$ \{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G $$ ...
Niall Taggart's user avatar
11 votes
1 answer
938 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 ...
D.S. Lipham's user avatar
  • 3,045
1 vote
0 answers
138 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 $\...
Meisam Soleimani Malekan's user avatar
8 votes
1 answer
360 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 ...
Pulcinella's user avatar
  • 5,507
4 votes
0 answers
101 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 ...
Luke Elliott's user avatar
10 votes
1 answer
222 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,...
Yann Peresse's user avatar
2 votes
1 answer
143 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?
TJP's user avatar
  • 51
3 votes
1 answer
132 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 ...
Pedro A. Matos's user avatar
7 votes
1 answer
224 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,...
Jens Bossaert's user avatar
0 votes
0 answers
135 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 ...
Taras Banakh's user avatar
  • 40.7k
2 votes
0 answers
77 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 (...
ABIM's user avatar
  • 4,989
1 vote
1 answer
230 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 ...
ABIM's user avatar
  • 4,989
3 votes
1 answer
811 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 ...
Alireza Abdollahi's user avatar
4 votes
2 answers
594 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 $E\...
Gabriel Medina's user avatar
2 votes
0 answers
86 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 ...
David Epstein's user avatar
1 vote
0 answers
139 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 ...
David Epstein's user avatar
7 votes
1 answer
481 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{\...
M. Winter's user avatar
  • 12.5k
2 votes
0 answers
179 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 ...
Jackson Morrow's user avatar
2 votes
0 answers
141 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 (...
Matthew Daws's user avatar
  • 18.5k
7 votes
1 answer
248 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 ...
Sam Hughes's user avatar
5 votes
2 answers
526 views

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. ...
Pedro A. Matos's user avatar
-2 votes
1 answer
128 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?
math112358's user avatar
4 votes
1 answer
130 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 ...
James Hanson's user avatar
  • 10.3k
1 vote
0 answers
128 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 $U(H)$ ...
Hugo Chapdelaine's user avatar
5 votes
1 answer
298 views

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 $G$ ...
Jens Bossaert's user avatar
3 votes
0 answers
156 views

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 $f \in L^2(G)$ ...
Constantin K's user avatar
1 vote
0 answers
241 views

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 ...
Ami's user avatar
  • 332
2 votes
1 answer
182 views

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,...
Ami's user avatar
  • 332
2 votes
1 answer
80 views

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, ...
Yemon Choi's user avatar
  • 25.5k
3 votes
0 answers
70 views

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 ...
John Coleman's user avatar
10 votes
1 answer
182 views

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 ...
Taras Banakh's user avatar
  • 40.7k
5 votes
0 answers
183 views

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 ...
D_S's user avatar
  • 6,100
2 votes
1 answer
201 views

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 ...
Ami's user avatar
  • 332
1 vote
0 answers
140 views

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, ...
Max Schattman's user avatar
5 votes
0 answers
291 views

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 ...
Jackson Morrow's user avatar
3 votes
1 answer
266 views

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-...
Ali Taghavi's user avatar
1 vote
1 answer
89 views

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 ...
Xqm's user avatar
  • 13
6 votes
0 answers
47 views

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 ...
geodude's user avatar
  • 2,129
2 votes
0 answers
56 views

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, ...
Duchamp Gérard H. E.'s user avatar

1
3 4
5
6 7
15