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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4
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1answer
144 views

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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Restrictions on pointed lifts of isometries

Let $M$ be a closed Riemannian manifold and let $f$ be an isometry of $M$ that fixes a point $\ast \in M$ and acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ ...
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1answer
242 views

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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25 views

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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1answer
245 views

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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0answers
67 views

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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60 views

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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1answer
109 views

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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1answer
452 views

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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157 views

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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1answer
218 views

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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237 views

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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130 views

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 ...
8
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1answer
166 views

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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1answer
335 views

About locally compact groups without compact subgroups

Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
9
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2answers
497 views

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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1answer
155 views

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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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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1answer
221 views

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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3answers
222 views

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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1answer
165 views

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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0answers
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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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1answer
64 views

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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0answers
120 views

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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0answers
143 views

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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110 views

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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0answers
25 views

Are there sequential paratopological groups of intermediate sequential orders?

Recall that a topological space $X$ is said to be sequential if a subset $A\subseteq X$ is closed if and only if it contains the limits of all convergent sequences which it contains. The definition ...
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0answers
171 views

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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1answer
339 views

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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58 views

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{...
7
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1answer
101 views

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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0answers
83 views

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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0answers
193 views

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 $F\in\mathbb{C}[x_1,\ldots, x_n]$ ...
4
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2answers
197 views

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 $$\int_G f(g) \mu(dg)$$ ...
3
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2answers
91 views

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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0answers
163 views

Does every topological group embed as a closed subgroup in an amenable group?

It is a standard result that closed subgroups of locally compact amenable groups are themselves amenable, so for example $F_2$, the free group on two generators, cannot be embedded as a closed ...
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94 views

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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0answers
142 views

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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2answers
218 views

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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0answers
72 views

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 (...
3
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1answer
223 views

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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0answers
59 views

How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
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1answer
133 views

Integration theory for functions and values with values in topological rings

I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure ...
7
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1answer
185 views

A group where the Weil topology induced by the Haar measure does not coincide with the original topology

Let $(G,\tau)$ be a locally compact Hausdorff topological group that is $\sigma$-finite with respect to the Haar measure $\mu:\mathcal{B}(G)\to[0,\infty]$ ($\mathcal{B}(G)$ is the Borel $\sigma$-...
13
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1answer
517 views

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 ...
5
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0answers
117 views

Matrix groups with two generators

Given two matrices $A,B\in{\rm{SL}}_2(\Bbb{R})$, is there any criterion guaranteeing that the subgroup they generate is discrete? What if one puts restrictions on $A,B$ e.g. they are both elliptic? ...
5
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2answers
133 views

Local cross-sections for free actions of finite groups

Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
2
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0answers
71 views

Constructing representations of a topological group from characteristic polynomials of a generating set

Given a topological group $G$ and a subset $S$ of $G$ that topologically generates it, what are the conditions under which an $n$-dimensional continuous linear representation of $G$ over an ...
2
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0answers
110 views

Galois representation absolutely irreducible after restricting to open subgroup of finite index

Let $E$ and $F$ be finite extensions of $\mathbb{Q}_p$. Let $\phi:\mathrm{Gal}(\overline{E}/E)\to GL_n(F)$ be an absolutely irreducible continuous representation. Assume that the restriction of $\phi$ ...

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