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2
votes
0answers
67 views
+50

If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable? Pietro Majer ...
3
votes
1answer
83 views

If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
6
votes
1answer
220 views

discrete group cohomology vs continuous group cohomology for profinite groups

Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map ...
2
votes
0answers
68 views

Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space. We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
1
vote
0answers
107 views

Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...
2
votes
1answer
77 views

Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
2
votes
0answers
90 views

Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$

Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$. Are there group topologies $\mathcal A$ ...
3
votes
1answer
87 views

algebraic groups over non-archimedean local fields acting on buildings

I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and ...
0
votes
0answers
54 views

Showing that a particular function from a locally compact Hausdorff group $ G $ to a $ C^{*} $-algebra $ A $ is Bochner-measurable

Suppose that we have the following data: A $ C^{*} $-algebra $ A $. A locally compact Hausdorff group $ G $. A strongly Borel mapping $ \alpha: G \to \text{Aut}(A) $, the automorphism group of $ A ...
1
vote
1answer
83 views

query about quasi-simple algebraic groups over local fields

Suppose that $G$ is an absolutely quasi-simple algebraic group over a non-archimedean local field $k$ (of either zero or positive characteristic). Is it known whether or not it is necessarily the case ...
8
votes
0answers
671 views

Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$. It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1]. Is $(\mathcal L,\subseteq)$ distributive? $$~$$ ...
7
votes
1answer
416 views

Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$

Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function $$f:G\times G\to G$$ $$f(x,y)=xy^{-1}$$ is continuous at $(1,1)$?
4
votes
1answer
169 views

Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point? The question is inspired by and related to ...
8
votes
0answers
404 views

A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{trivial}, \mathcal T_{discrete}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with $$\mathcal ...
0
votes
1answer
52 views

Parallel group topologies on Prüfer groups

Let $p$ be a prome number. Are there group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that $$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$
4
votes
1answer
111 views

Equations and random subgroups in compact groups

EDIT: Here is a more specific question. Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...
4
votes
0answers
78 views

Is every compact monothetic group metrizable?

If $G$ is a compact (Hausdorff) topological group with a dense cyclic subgroup, is it necessarily true that $G$ is first countable? This claim seems to be implicit in a paper that I am reading at the ...
5
votes
0answers
56 views

Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$. ...
3
votes
1answer
100 views

Infimum of two group topologies

Let $G$ be a non-abelian group and $\mathcal S$ and $\mathcal T$ be group topologies on $G$. What is the largest group topology $\tau$ on $G$ with $\tau \subseteq \mathcal T\cap \mathcal S$? In ...
5
votes
1answer
112 views

Is an extension of compact Hausdorff topological groups compact?

Let $1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1$ be a short exact sequence of topological groups (i.e., all maps are continuous, $A = \mathrm{Ker}(c)$, and $C = ...
6
votes
2answers
222 views

Union of conjugates of a closed subgroup of a compact group

Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$. Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of ...
0
votes
0answers
45 views

The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true? In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...
2
votes
0answers
71 views

Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as ...
2
votes
0answers
317 views

Topological proof of a result in Logic

I proved the result below using logic. My questions: Can this theorem be proved by purely topological means? Do you know any theorems that either can be used to prove the same result, or which give ...
0
votes
1answer
117 views

Simple but topologizable [closed]

Do you have an example of an infinite simple group with at least 3 distinct group topologies on it‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
1
vote
2answers
221 views

Locally compact vs. compactly generated in group theory

I'm studying the Creutz-Shalom paper on the normal subgroup theorem for dense commensurators of lattices. The authors state their results for locally compact and compactly generated groups. I do not ...
7
votes
1answer
367 views

Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
3
votes
2answers
245 views

The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group? (My ...
14
votes
1answer
358 views

In what topological abelian groups does convergence to zero imply summability?

(This question has been on math.SE for over a week and has not gotten any answers.) Let $G$ be a (T$_0$) topological abelian group, and let $0$ be its identity element. Assume that for all index ...
10
votes
1answer
230 views

Compact Quantum Groups and the Existence of the Classical Haar Measure

Before I state my question, let me provide the definition of a compact quantum group. Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if ...
9
votes
0answers
135 views

Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
2
votes
0answers
82 views

Free profinite completions

Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?
3
votes
0answers
94 views

Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?
4
votes
3answers
529 views

Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
4
votes
0answers
64 views

Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...
5
votes
1answer
103 views

When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube

Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus ...
11
votes
2answers
397 views

rationalization of classifying spaces

This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway: Let $G$ be a simply-connected topological group. In particular, it is an ...
6
votes
1answer
212 views

Topological groups defined by completely disconnected subgroups

Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
4
votes
1answer
112 views

Automorphism group of compact abelian group

I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as ...
1
vote
2answers
144 views

Hausdorffness inheritance in topological groups

Suppose $\mathcal T$ and $\mathcal S$ are two compatible Hausdorff topologies on a group $G$ and $\mathcal R$ is a maximum compatible topology on $G$ with $\mathcal R \subseteq \mathcal T\cap \mathcal ...
0
votes
0answers
38 views

Semi-continuity of injectivity radius

Let $H$ be a locally compact second countable topological group and $X$ be a locally compact second countable space. Assume that $H$ acts continuously on $X$ and that for every $x$, $H_x=\{h \in H ; ...
2
votes
1answer
111 views

Connectedness properties of groups of homeomorphisms

Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...
2
votes
2answers
167 views

injective implies completion injective?

Background and definitions. Let $k$ denote a field complete with respect to a non-trivial non-archimedean norm. Let $R$ be the integers in $k$, and say $\pi\in R$ with $0<|\pi|<1$ ($\pi$ ...
5
votes
1answer
263 views

Locally finite compact groups

I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...
0
votes
1answer
80 views

Is a weakly separable group always Lindelöf?

By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
11
votes
2answers
414 views

Restriction of “$\pi_{1}$” to topological groups

Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$?
15
votes
1answer
488 views

Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
1
vote
0answers
55 views

Continuity of inversion and composition in certain topological groups

For $k\in\Bbb{N}$ ($k\geqslant 1$) and $\alpha\in]0,1]$, let $H_{k,\alpha}([0,1])$ be the group of orientation preserving $C^{k,\alpha}$ diffeomorphisms of the closed unit interval $[0,1]$. We furnish ...
2
votes
2answers
311 views

Is any continuous group homomorphism from R to C* an exponential map? [closed]

Consider $\mathbb{R}$ to be an additive topological group, and $\mathbb{C}^{\ast}$ to be a multiplicative topological group. Is the following statement true? If so, then how can one prove it? ...
4
votes
2answers
339 views

Irreducible representations of compact groups

Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ...