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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Digesting a proof of Bieberbach's theorem on crystallographic groups

Here is a proof of Bieberbach's theorem: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.1436&rep=rep1&type=pdf I can follow the computing details, but I have no intuitive sense ...
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how many group structures make $S^1$ a topological group?

let $S^1$ be the subspace of $R^2$ given the usual topology. How many group structures make $S^1$ a topological group?
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176 views

Is the action of a finite group always discontinuous?

Let $G$ be a finite group acting on a topological space $X$. Is it true that the action of $G$ is always proper? To say that it is proper we have to show that the map $\theta : G \times X \...
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A problem of a discrete group of smooth isometries acting discontinuously on a smooth manifold.

Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly ...
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39 views

A specific problem on locally compact topological group Q and non existence of Haar measure

I have recently taken a course on topological groups and their Haar measure and I should first mention I am still a beginner so even though I thought about this for a while, I was hoping someone here ...
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1answer
63 views

Topology of SL(2,R)

I am interested in visualizing the Lie group $SL(2,\mathbb R)$ topologically. Writing a matrix as $A = \begin{pmatrix} x+w & -y+z\\ y+z& x-w \end {pmatrix}$ I could identify a matrix with ...
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1answer
23 views

If $G\curvearrowright X$ and $H\leq G$ then $\bar{H}x = \overline{Hx}$

If $G$ is a topological group acting effectively on a topological space $X$ and $H$ is a subgroup of $G$ then is it true that $\overline{H}x = \overline{Hx}$, where $\overline{H}$ is the closure of $H$...
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56 views

Compatibility of group structure and topological structure for topological groups

I am fairly new to the concept of topological groups, and would like to understand the underlying idea. My question is about the compatibility between the two structures. The definition of a ...
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47 views

Closed subgroups of $Z_{p}^{\times}$

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is $...
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$(\mathbb{C}^n -\{ 0 \}) / \mathbb{Z}_k$ [duplicate]

I would like to know what the space $(\mathbb{C}^n -\{ 0 \}) / \mathbb{Z}_k$ looks like topologically and geometrically, where the group action is given by $z \mapsto e^{\frac{2 \pi i}{k}} z$. For n=1 ...
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81 views

The existence of a limit point of a closed set

Walter Rudin define a closed set as: 2.18 (d) A subset E of a metric space is closed if every limit point of E is a point of E. I don't see in this definition nothing about the existence of a ...
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85 views

Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$

I would like to know topologically what the space $(\mathbb{C}^n - \{0\})/\mathbb{Z}_k$ may be thought of as. The paper I am reading says that we let $\mathbb{Z}_k$ act on $\mathbb{C}^n - \{0\}$ via ...
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1answer
47 views

Discrete VS finite groups

I am having some troubles to understand what the difference is between discrete and finite groups. I know they are defined differently, I can't quite understand the difference. I am guessing every ...
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2answers
25 views

Is there such thing as an $n$-fold cover of $SO(n)$?

I know already that the $Spin(n)$ group is a double cover of $SO(n)$, but is there such a thing as a triple cover, fourfold or $n$-fold cover? I am interested to know if there are 3-spinors (?) or ...
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66 views

Real orthogonal matrices with fixed entry value has zero Haar-measure?

Consider the set of real orthogonal matrices of size $n \times n$ such that one entry, say $a_{i,j}$ for fixed $i$ and $j$, satisfies $a_{i,j}=0$. Has this set zero Haar-measure? A simple proof, in ...
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34 views

Classification of symmetric space of non compact type

Is there a classification of rank one symmetric space of non compact type ? Remark, for rank 1, There are three families : 1) hyperbolic n-space, corresponding to the Lie group SO(n,1). 2) complex ...
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31 views

Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
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1answer
54 views

Absolute Galois group and continuous homomorphism.

Let $G=\mathrm{Gal}(\bar K/K)$ be the absolute Galois group of a number field $K$. I have some problem to understand that the homomoprhism $$ \rho :G \longrightarrow \mathrm{GL}_r(E)$$ $E$ a finite ...
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20 views

What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
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54 views

Examples of left-topological compact semigroups

I was reading chapter in Todorčević's book Topics in Topology (LNM 1652, DOI: 10.1007/BFb0096295) which deals with the semigroup $\beta\mathbb N$. Several results about left-topological compact ...
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38 views

If a group and a groupoid have the same profinite completions, can anything be said about them?

Pretty much what I ask in the title. Suppose $\mathbb{G}$ is a group and $\mathcal{G}$ is a groupoid. Consider their profinite completions $\overline{\mathbb{G}}$ and $\overline{\mathcal{G}}$. If ...
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1answer
33 views

Is there an relation between the notions of continuous and discontinuous group actions?

Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if $G_x :=$ the stabilizer of $x$, is finite. $x$ has an ...
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46 views

Continuity using topological groups

I have been give this question. I am just a bit confused where to start? Prove that $g : \mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\to \mathrm{GL}(n,\mathbb R)$ given by $g(x, y) := xy$ (...
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1answer
18 views

Compact subgroups of a topological group

Consider $G=(0,\infty )$ , with the metric induced by Euclidean metric from $\mathbb R$ .$G$ is a group under multiplication . Then which subgroups of it are compact $?$ Now $G=\...
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32 views

What does the topology on $SL(n, \mathbb{R}) / SL(n, \mathbb{Z})$ intuitively look like?

I have come across Mahler's compactness criterion, and am having trouble wrapping my head around the topology of the moduli space of unit volume lattices. Is there an intuitive way to think about it, ...
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1answer
29 views

Neighborhood base at the unit element in a topological group

One can define the open set $U$ be such that $U$ is open if for every $x\in U$, $xV\subset U$ for some $V\in\mathcal{N}$. (2) and (3) would give the desired continuity for group multiplication and ...
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28 views

Proving the Existence of n-linked knots

I was reading up on knots and links and came across: The Hopf Link: https://en.wikipedia.org/wiki/Hopf_link Solomon's Knot (Double Link): https://en.wikipedia.org/wiki/Solomon%27s_knot Which got me ...
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1answer
42 views

Topological groups, is continuity a requirement or consequence of this definition?

On the Wikipedia-page for topological groups, it is stated as a requirement that the binary operation and the inverse operation is continuous. In Munkres he makes this definition: A topological ...
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27 views

Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on $N(...
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61 views

Fubini's theorem for topological monoids

A topological commutative monoid is a commutative monoid $(X,+)$ with a topology $\cal O$ such that $+$ is a continuous function. There is a notion of infinite sum on topological commutative monoids, ...
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33 views

How does one visualize elements of standard Iwasawa algebra?

How does one think of elements of standard Iwasawa algebra of a profinite abelian group $B$ ? I mean like elements of $Z_{p}$ are represented as infinite series, is there a way to think of elements of ...
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1answer
131 views

Is $1+T$ a topological generator for $Z_{p}[[T]]$?

Is $1+T$ a topological generator for $Z_{p}[[T]]$? ($Z_p$ is the ring of p-adic integers)
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1answer
46 views

Classifying topologically cyclic groups

The group of p-adic integers has a dense cyclic subgroup, i.e. it is topologically cyclic What are some (or all) other non-trivial (i.e. non-discrete) examples of such groups?
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93 views

Topological group, a big exercise!

Let $(G,d)$ be a group that is also a compact metric space, and G is topological (The maps $G \times G \to G, (x,y) \to xy$ and $G \to G, x \to x^{-1}$ are continuous). And then we consider $(G,\rho)$,...
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Smooth admissible representations, Hom, tensor and extension of scalars.

Let $G$ be a locally profinite group, and consider $V$ and $W$ smooth admissible representations of $G$ over some field $F$ (or char. $0$). Let $E/F$ be any field extension. I'd like to find ...
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1answer
142 views

Can there be a Diaconis-Shahshahani Upper Bound Lemma for Compact Groups?

Let $G$ be a finite group and $\nu\in M_p(G)\subset \mathbb{C} G$ a probability measure on $G$ and let $\pi$ be the uniform distribution on $G$. Denote by $d_\rho$ the dimension of a non-trivial ...
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Is $(X_G, d_G)$ a compact manifold?

Let $G$ a compact topological group act on $(X,d)$ by isometries. We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It ...
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1answer
48 views

Definition of Roelcke uniformity

Let $(G,\mathcal T)$ be a topological group. The set of all uniformities on $G$ forms a lattice $\frak A$ and the set of all uniformities on $G$ producing $\cal T$, forms a lattice $\frak B$. The ...
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106 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...
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31 views

Explicit construction of Haar measure on a locally profinite group

Let $G$ be a locally profinite group. A Haar measure $\mu$ on $G$ is a measure defined on the $\sigma$-algebra $\mathcal B(G)$ of all Borel sets of $G$ with the following properties. 1) $\mu(K) \...
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1answer
36 views

Two disjoint compact sets in a topological group

Let $(G, \cdot )$ be a compact (Hausdorff) topological group. If $A$ and $B$ are two disjoint compact subsets of $G$, how can we show that there exists a nonempty open set $V$ such $A\cdot \overline{V}...
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118 views

Explicit construction of Haar measure on a profinite group

Let $G$ be a profinite group. It is known that in $G$, every neighborhood of the identity element contains an open compact subgroup. I would like to explicitly construct the Haar measure on $G$. The ...
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1answer
78 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
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Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then $\...
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1answer
107 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
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1answer
23 views

equality in uniform space and topological groups

I wanted to ask the following: If I have a topological group $G$, I know I can create a base for a uniform space as follows: for each $U$ a neighborhood of e, we define $V_u= \{(x,y):x^{-1}y\in U\}$. ...
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1answer
58 views

Topology question with closed sets.

Let $ K\subseteq \mathbb{R}^n$ be a compact set and let $E\subseteq \mathbb{R}^n$ be a closed set. ***Its also given that $ \inf \{d(x,y)|x\in K, y\in E\}=0$. $ d(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$ ...
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54 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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Amenability; topology on power sets

I am currently reading the proof of Proposition 2.2. (a direct limit of discrete amenable groups is amenable) in http://people.maths.ox.ac.uk/kar/amenable.pdf . The proof uses the Tychonoff theorem ...
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1answer
62 views

Symmetric open neighborhood in topological group

I want to prove that ($\ast$) If $G$ is a Hausdorff topological group, then for every neighbourhood $U$ of $e$, there exists a symmetric neighbourhood $V$ of $e$ ($V^{-1}=V$), s.t $V*V \subset U$ ...