**33**

votes

**4**answers

4k views

### Fundamental group as topological group

Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...

**7**

votes

**1**answer

217 views

### Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations.
Theorem The normal subgroups of $S_\infty$ are ...

**0**

votes

**2**answers

359 views

### Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...

**4**

votes

**0**answers

103 views

### Automorphism group of a structure without the SAP

A few years ago, a number of examples were given of Fraisse structures without the SAP in answer to the question raised in A Fraïssé class without the strong amalgamation property.
It is ...

**1**

vote

**0**answers

65 views

### Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...

**0**

votes

**0**answers

74 views

### Right split for homomorphism onto $S_\infty$

Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...

**2**

votes

**0**answers

87 views

### Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?

Recall that a locally compact group $G$ is said to be
an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure;
an $[SIN]$ group, if each neighborhood of the identity includes a ...

**4**

votes

**1**answer

230 views

### Weil's Haar measure construction from below

Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...

**8**

votes

**1**answer

174 views

### Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...

**3**

votes

**0**answers

55 views

### Characteristically simple locally compact abelian groups

Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `...

**5**

votes

**1**answer

74 views

### Is the Hofer topology second countable?

Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}^c(M,\omega)$ denote the group of compactly supported Hamiltonian diffeomorphisms of $(M,\omega)$. Is the Hofer topology on $\...

**4**

votes

**0**answers

126 views

### LCH topologies on Groups that are not group topologies

Ellis's 1957 paper on Locally Compact Transformation groups proves the following:
A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately)...

**0**

votes

**0**answers

54 views

### A question about Raikov complete topological groups

I asked this yesterday on Math Stackexchange, but didn't get any comments. So I thought I might ask here too.
Let $G$ be a topological group and let $G^*$ be its Raikov completion, i.e its completion ...

**1**

vote

**1**answer

61 views

### Set product of profinite subgroups of a compact group is profinite

Let G be a compact group, suppose $G=AB$ where $A$,$B$ are profinite subgroups of $G$. Is it true that G is profinite group?

**13**

votes

**0**answers

445 views

### A possible mistake in Walter Rudin, “Fourier analysis on groups”

I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):
Suppose $E$ is a coset in $\Gamma_2$ ...

**7**

votes

**2**answers

1k views

### Haar measure for large locally compact groups

In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a ...

**6**

votes

**2**answers

196 views

### A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable,
locally connected and has finite topological dimension, yet fails
to be locally compact?

**2**

votes

**0**answers

87 views

### Centralizer of a dense subgroup in a maximal subgroup of a reductive group

I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...

**1**

vote

**1**answer

170 views

### Every norm-decreasing algebra morphism $L_1(G)\to\mathcal{B}(E)$ comes from a group representation

In section 8 of this paper http://arxiv.org/abs/math/0611833v3 the author proves the following:
If $E$ is a reflexive Banach space, $G$ a locally compact group and $\pi:L_1(G)\to\mathcal{B}(E)$ a norm-...

**3**

votes

**1**answer

104 views

### Why are the convolvers in the bicommutant of the pseudo-measures? ($CV_p(G)\subseteq PM_p(G)''$)

Let $G$ be a locally compact group. For $1<p<\infty$ let $\lambda_p:G\to\mathcal{B}(L^p(G))$ (resp. $\rho_p:G\to\mathcal{B}(L^p(G))$) be the left (resp. right) regular representation. $CV_p(G)$ ...

**4**

votes

**1**answer

158 views

### Discrete spectrum and almost periodicity

According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to \...

**3**

votes

**0**answers

59 views

### Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...

**9**

votes

**6**answers

2k views

### What is a good book on topological groups?

I am looking for a good book on Topological Groups. I have read Pontryagin myself, and I looked some other in the library but they all seem to go in length into some esoteric topics.
I would love ...

**4**

votes

**1**answer

200 views

### The fibration map $Diff(M) \rightarrow Emb(N,M)$

Let $M$ be a non-compact manifold, equipped with a (closed?) submanifold $N\subset M$. The action of $Diff(M)$ on the set of embeddings $N\hookrightarrow M$ induces a map
$$
Diff(M) \rightarrow Emb(N,...

**18**

votes

**2**answers

1k views

### Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...

**1**

vote

**0**answers

77 views

### Subgroups of finite non-zero Haar measure of abelian locally compact groups

Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.

**2**

votes

**1**answer

114 views

### Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously

My question will rely on a clarification of a proof, which I simply don't understand.
Let us denote by $X$ a pseudo-riemannian symmetric space and define
$$
Z_{\mathrm{Iso}\left(X\right)}G(X) = \{\, ...

**3**

votes

**0**answers

58 views

### Kind of multiplicative total boundedness in Hausdorff compact rings

Let $(R,\cal T)$ be a unital Hausdorff compact topological ring and let $A$ be an open subset of $R$ containing $1$. Is there a finite set $B$ with $AB=R$?

**6**

votes

**0**answers

216 views

### Two different Thom diagonals in recent literature?

Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...

**1**

vote

**0**answers

70 views

### Not normal connected component of a right topological group

Let $\cal T$ be a locally compact topology on a group $G$ and $(x,y)\mapsto xy^{-1}$ be continuous at $(1,1)$ and for every $a\in G$, $x\mapsto xa$ be continuous everywhere with respect to $\cal T$.
...

**13**

votes

**8**answers

3k views

### Haar measure on a quotient, References for.

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz).
Right from the very beginning, Weil uses the ...

**4**

votes

**0**answers

123 views

### When can a locally compact group be approximated by discrete subgroups?

This question is about partitioning a (locally) compact group into cells by using discrete subgroups.
Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a ...

**2**

votes

**2**answers

256 views

### Abelian extremely amenable group?

Is there a nontrivial commutative Hausdorff topological group that is extremely amenable?
Recall that a topological group is called extremely amenable if any continuous action on a compact Hausdorff ...

**6**

votes

**0**answers

214 views

### Is $k(\!(x,y)\!)$ a topological field?

More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...

**5**

votes

**0**answers

59 views

### Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups
in the topological setting, I face the following problem.
Let $G$ be a locally compact amenable ...

**12**

votes

**0**answers

232 views

### A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...

**2**

votes

**0**answers

71 views

### Distal actions on coset spaces

Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point ...

**11**

votes

**1**answer

248 views

### A generalization of residual finiteness to topological groups

Consider the following generalization of residual finiteness to
topological groups.
A locally compact Hausdorff group $G$ is called residually compact if
for every compact $K \subseteq G$ there is a ...

**2**

votes

**1**answer

178 views

### Can we Characterise Rings of Continuous Functions?

Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...

**0**

votes

**0**answers

49 views

### Left introversion operators associated to function spaces on semigroups

I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...

**6**

votes

**4**answers

918 views

### Topological structure of SO(n) as a product

I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product
$$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$
Allen Hatcher [1, p. 293 f.] ...

**5**

votes

**1**answer

184 views

### The evaluation fibration of a transitive, effective topological group action

Does anybody know a reference to the following fact?
If $G$ is a topological group acting transitively and effectively on a space $X$, then the evaluation map $G \rightarrow X$, $g \mapsto g \cdot ...

**17**

votes

**0**answers

432 views

### The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...

**25**

votes

**1**answer

737 views

### Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$:
$A' = aA$,
$B' = bB$.
Suppose it is known that $...

**29**

votes

**1**answer

1k views

### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$...

**1**

vote

**0**answers

75 views

### Topologically finitely generated residually finite group

Suppose that $G$ is a topologically finitely generated profinite group,
and let $H$ be a subgroup of countably infinite index. Is $H$ necessarily topologically finitely generated with the subspace ...

**3**

votes

**1**answer

122 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{...

**6**

votes

**1**answer

95 views

### Can each non-open analytic subgroup of a Polish abelian group be covered by countably many closed Haar null subsets?

By a result of Laczkovich ('Analytic subgroups of the reals' Proc AMS Vol 126 (1998)), any non-open analytic subgroup of a Polish locally compact group can be covered by countably many closed Haar ...

**4**

votes

**0**answers

2k views

### Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups

Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?

**2**

votes

**1**answer

93 views

### Just-not-nilpotent-by-compact quotient of a locally compact group

It is known (and not complicated to prove) that for a finitely generated not virtually nilpotent group $G$, we can pass to a quotient $G/N$ of $G$, such that the quotient is just not virtually ...