# Questions tagged [topological-groups]

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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### Does the compact-open topology retain topological groups?

Let $X$ be a topological space and $Y$ a topological group. Then $C(X,Y)$ is a group, and can also be endowed with the compact-open topology. Is $C(X,Y)$ in the compact-open topology necessarily a ...
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### Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?

Let $G$ be a compact Lie group. An Abelian Lie subgroup $A \leq G$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $A'$ such that $A \leq A' \leq G$, then $A' = A$. Of course any ...
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### Continuous surjection from $X(D_n)$ onto $\operatorname{Homeo}_0(D_n)$

Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...
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### Reference request: placing a set with respect to the integer grid

For $x=(x_1,...,x_n)\in \mathbb{R}^n$, let $Q_x=(x_1,x_1+1)\times ...\times (x_n,x_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft here) the following ...
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### Can you give me an example of a totally disconnected subgroup of a topological group that is not a topological group? [closed]

In the book The topology of fiber bundles, Steenrod characterize bundle over a base $X$ and totally disconnected structural group $G$ as follows. Theorem: Let $X$ be arcwise connected, arcwise ...
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### Locally compact Polish groups acting on standard Lebesgue spaces

If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...
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### Seeking to understand meaning of “von Neumann spectrum” in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page) I find that towards the end of the ...
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### Residually finite group with dense finite index subgroups

Let $G$ be a locally compact Hausdorff topological group whose underlying abstract group is residually finite. Let $H\subset G$ denote the intersection of all finite-index, closed subgroups. Is there ...
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### Is there an orbit map without path lifting property?

I am looking for an example of a topological group $G$ acting by homeomorphisms on a metrizable space $X$ such that the orbit map $X\to X/G$ doesn't have the path lifting property, that is, there is a ...
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### Quantum analogue of certain property of compact groups

Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$. What is a precise description of a maximal ,or in some sense ...
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### Does a cocompact subgroup of a topological group contain a cocompact normal subgroup?

Motivation: It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$. Our question: Let $G$ be a topological ...
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### Empty interior of union of cosets?

The following question arises from trying to understand Lemma 1.3(ii) of arXiv:math/0405063. I believe a particular case of the proof (and in fact I think the proof is essentially equivalent to this ...
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### Homomorphism defined on a dense subset extending in different way with respect to different target topologies

Let $G$ be a profinite group with a dense subset $S\subset G$. Let $H_1$ and $H_2$ be two topological groups whose underlying abstract group is the same. Can there exist homomorphisms of topological ...
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### Nilpotency of topological groups

A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups $$\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$$ ...
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### Why are homeomorphism groups important?

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
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### Homomorphisms from circle to $GL(k,\mathbb{R})$ [duplicate]

Example 3 at the website tricki proves that every measurable homomorphism of groups from the circle to the non-zero complex numbers is continuous. Is there any analogous (true) statement for ...
### Continuous vs $L^2$ homomorphisms from circle to non-zero complex numbers
Let $T:S^1\to C^\ast$ be a group theoretic homomorphism from the circle to the non-zero complex numbers. Presumably it is true that if $T$ is $L^2$, then it is continuous. Is there a simple proof, or ...