A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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When a group acts on a metric space isometrically, transitively, and faithfully, is it a closed subspace of the isometry group?

Suppose you have a group $G$ acting on $ (M,d)$ a compact metric space by isometries (meaning $d(gx,gy) = d(x,y)$ for all $x,y \in M$ and all $g \in G$), transitively and faithfully. You can define ...
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1answer
21 views

Is there a natural (non-trivial) topology for the automorphism group of a locally compact abelian group?

I've been thinking about automorphism groups of locally compact abelian groups somewhat over the last few days. It's not hard to see that in the case of the torus and integers, the (continuous) ...
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0answers
29 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
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14 views

Is this a compact group?

Consider $$x(t)=e^{-iHt}x(0)$$ and define $$G=\{ e^{-iHt}\mid t\in \mathbb{R}_{\geq 0}\}$$ Also write $\bar{G}$ to the closure of $G$ wrt the Euclidean topology. Q: is $\bar{G}$ a compact group? ...
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1answer
76 views

What is a Topological Group Intuitively?

What is a topological group intuitively, beyond just being able to say things are near to each other in a group, and why is it a good idea to consider this theory as part of general topology as ...
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2answers
65 views

What are some compact (Hausdorff) groups?

I just realized today that I don't know any compact groups that aren't profinite groups or Lie groups. Generalizing from these, a product of compact groups is again a compact group, a closed ...
2
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1answer
27 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
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2answers
60 views

Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
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0answers
40 views

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff.

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff. I am working through some notes on Geometric Group Theory and I am having a hard time ...
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1answer
48 views

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash ...
4
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1answer
54 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
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1answer
61 views

Equivalent definitions of a lattice in a real vector space of finite dimension

I'm currently trying to work my way through chapter seven of Serre's book "A Course in Arithmetic" with a view to learning about modular forms. During the course of this chapter the book begins to ...
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1answer
18 views

Closed kernel in a compact group is open

The way I think it should work is that $${\rm ker} = \bigcap_{g \notin {\rm ker}} (G - g\,{\rm ker}),$$ with each $G - g\,{\rm ker}$ open. Since $G$ is compact, there should, in fact, only be ...
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0answers
44 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
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1answer
40 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
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1answer
41 views

Characters of a Group: two definitions

If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms: $$\chi:G\longrightarrow\mathbb ...
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1answer
25 views

Product of compact subsets of a topological group is compact

I need to show that if $G$ is a topological group and $A,B\subseteq G$ are compact subsets, then $AB$ is compact. Thanks
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0answers
33 views

Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
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1answer
51 views

Topological group with discrete topology

Let $G$ be a topological group. I came to know that if I can show the existence of a homeomorphism of $G$ which moves only finitely many points of $G$, then $G$ has only discrete topology. How can I ...
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2answers
115 views

Creating a Topological group from modulo multiplication Group.

If I were to create a Topology out of the Modulo 3 Multiplication group $\mathbb{Z}_3$, what elements would it consist of and why? So $\mathbb{Z}_3 = \{0,1,2\}$ as a group over modulo 3. What are the ...
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0answers
71 views

Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
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0answers
29 views

Can a compact topological group have the trivial topology?

I want to show that the following are equivalent for a compact topological group $G$: $G$ is the inverse limit of finite groups $G_i$. There's a family $\left\{N_i\right\}$ of open normal subgroups ...
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1answer
66 views

Locally compact groups

Let $S= G_1\bigcup G_2 $, where $G_1$ and $G_2$ are two groups. If $S$ is locally compact, is it true that either $G_1$ or $G_2$ is locally compact? Thank you.
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60 views

Any topological group is Hausdorff if $\{1\}$ is closed.

Let $G$ be a topological group and $\{1\}$ is closed where $1$ is the identity element of $G$. Then for any two distinct element $x,y \in G$ , we have an open neighbourhood $U$ of $x$ not containing ...
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1answer
48 views

Is every compact monothetic group metrizable?

If $G$ is a compact (Hausdorff) topological group with a dense subgroup $H \cong \mathbb Z$, is it necessarily true that $G$ is first countable? This claim seems to be implicit in a paper that I am ...
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1answer
36 views

Proving closure of unit space of a Hausdorff groupoid

For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed. For topological groupoids, defined ...
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0answers
43 views

Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
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1answer
55 views

quotient of an amenable group

I have a question about amenable groups. The notion of amenability I am using is: The action of $G$ on $k$ ($k$ locally compact in a topological vector space) is amenable if there exists a point $x ...
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0answers
32 views

Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...
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0answers
20 views

Considering the right Haar measure on the affine group, how does the absolute value come in?

Let $ G $ be the affine group with group action defined by $ (b,a)\cdot(x,s) = (ax+b,as) $ then it is a locally compact group and as such has a Haar measure. In particular the left Haar measure is $ ...
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31 views

$\widehat{\mathbb{T}}$ can be identified with $\mathbb{Z}$

$ \mathbb{T} \stackrel{\text{def}}{=} \{ z \in \mathbb{C} : |z| = 1 \}$ $\widehat{\mathbb{T}} \stackrel{\text{def}}{=} \text{Hom}(\mathbb{T},\mathbb{T})$ To show that $\widehat{\mathbb{T}}$ can be ...
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1answer
68 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
2
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1answer
37 views

Are bounded subsets of Lie groups totally bounded

Let $ G $ be a finite dimensional real Lie group, and take a bounded ball $ B_R(e) \subset G $ in it, coming from the Riemannian metric, which itself is induced from an inner product on $ \mathfrak{g} ...
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Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
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2answers
51 views

Is intersection of connected subgroups connected?

Let $G$ be a compact group. If $A_{\alpha}$, $\alpha \in I$ is a family of closed connected subgroups in $G$, then is it true that $\bigcap_{\alpha \in I}A_{\alpha}$ is connected?
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26 views

Summand of a subgroup in a torus

Let $\mathbb{T}^n := \mathbb{R}^n / \mathbb{Z}^n$ denote the $n-$dimensional torus. If $K$ is a closed normal subgroup of $\mathbb{T}^n$, then does there exist a subgroup $L$ of $\mathbb{T}^n$ such ...
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112 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
6
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1answer
86 views

Profinite topology of a Group

Let $G$ be a group. Consider now the set of all (left for instance) cosets in $G$ of subgroups of finite index. This set is a base for a topology in $G$. I found somewhere that if $G$ is residually ...
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1answer
57 views

A paratopological group with intersection of neighborhoods of a point non-closed

Do you have an example of a (para)topological group $(G,\mathcal T)$ such that $\bigcap_{V\in \mathcal N_1} V$ is not closed? $\mathcal N _1$ is the set of all neighborbood of the identity element of ...
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0answers
71 views

Explanation of a passage in Atiyah / Macdonald

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact ...
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1answer
24 views

Continuity of Multiplication by Fixed Element

This is likely a simple question that I'm just missing, but nothing immediately came to mind. When dealing with topological monoids, it is necessary to prove that the group operation is continuous. ...
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2answers
54 views

Introduction to discrete subgroups of the euclidean group

I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space). Even though I searched quite a bit, I was unable to find a good ...
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0answers
35 views

A homeomorphism of topological groups.

Let $L/K$ be a Galois field extension and $(I, \leq)$ be the directed set of all finite Galois extensions $E$ of $K$ contained in $L$ (we say $E^{\prime}\leq E$ if $E^{\prime}\subseteq E$). If ...
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2answers
50 views

Showing that every path can be well-divided?

Let $\gamma: [0,1] \rightarrow S^1$ be a path. We'll say that $\gamma$ is well-divided if there are $a_1,...a_n$ such that: $a_1=0$, $a_n=1$ $\forall_{1\leq i < n}: a_i<a_{i+1}$ ...
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0answers
33 views

Continuous maps of topological groups

Suppose $\theta: G\to K$ is a homomorphism of topological groups. We know that $G$ and $K$ each have fundamental systems of open neighbourhoods of the identity. If I want to show that $\theta$ is ...
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2answers
135 views

What topological restrictions are there for a topological space to be a group?

I'm trying to provide a group structure for some Riemannian surfaces. I heard that the following result holds: Let $X$ be a compact Riemannian surface. Then $X$ admits a group structure if, and ...
0
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1answer
25 views

Haar measure of point sets

Let $G$ be a locally compact group with Haar measure $\mu$ (left or right doesn't matter to me). I know that the Haar measure is positive on open sets. What can be said about the Haar measure on ...
0
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1answer
30 views

Elements Outside the Identity Component $SO^+(1,\,3)$ of the Lorentz Group $O(1,\,3)$

I have been answering a question on Physics Stack exchange to do with the difference between the "proper orthochronous" (i.e. identity component of) the Lorentz group $SO^+(1,\,3)$ and the Lorentz ...
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2answers
73 views

Closure of a topological group

Let $G$ be a topological group. Define $H(g)=\{g^n\}^{\infty}_{n=-\infty}$ for each $g \in G$. I need to prove that the closure of the set $H(g)$ is a commutative subgroup of $G$. But I am not sure ...
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0answers
29 views

Connected component and group action

Let $G$ be a topological group acting on a set $X$. Let $x \in X$ and consider the orbit $G.x$ endowed with the topology coming from the quotient $G/ Stab(x)$. If $G^0$ is the connected component of ...