# 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).

587
questions

**4**

votes

**1**answer

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 ...

**3**

votes

**0**answers

80 views

### 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 $\tilde{f}$ ...

**8**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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.

**1**

vote

**1**answer

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 ...

**2**

votes

**0**answers

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 $\...

**0**

votes

**0**answers

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}...

**10**

votes

**0**answers

192 views

### 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 ...

**1**

vote

**1**answer

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(\...

**3**

votes

**1**answer

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}^\...

**5**

votes

**0**answers

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^{\...

**1**

vote

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**3**

votes

**0**answers

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**

votes

**1**answer

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,...

**8**

votes

**1**answer

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**

votes

**2**answers

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 ...

**6**

votes

**1**answer

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 ...

**5**

votes

**0**answers

47 views

### 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 $...

**1**

vote

**1**answer

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 ...

**9**

votes

**3**answers

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 ...

**2**

votes

**1**answer

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
...

**1**

vote

**0**answers

20 views

### 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 ...

**1**

vote

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**5**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**7**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**2**answers

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 ...

**9**

votes

**0**answers

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 ...

**6**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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 ...

**9**

votes

**2**answers

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 ...

**3**

votes

**0**answers

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**

votes

**1**answer

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 = \...

**2**

votes

**0**answers

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}...

**4**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 $\phi$ ...