# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $V*V \subset U$ ...
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### Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
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### Every representation of compact group is a direct sum of irreducible

Recently I asked about (references to) some results concerning representation theory of compact topological groups: here is the discussion Representation theory of locally compact groups In ...
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### Question regarding characters and point open topology2

this is a follow-up question for the following one: Dual group of G with point open topology is an intersection of C(G,T) and a closed set In the book of Banaszczyk - "Additive subgroups of ...
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### Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
### Set of discrete orbits under subgroup of $Isom(\mathbb{E}^n)$ is clopen
I'm trying to solve the following exercise (exercise 1.4 from Szczepanski's "Geometry of Crystallographic Groups"): Let $\Gamma$ be a subgroup of $I(\mathbb{E}^n)$, the group of isometries on ...