All Questions
Tagged with lie-groups topological-groups
79 questions
20
votes
2
answers
2k
views
Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?
The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...
- 2,974
17
votes
0
answers
1k
views
What groups are Lie groups?
We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
- 47.3k
12
votes
4
answers
2k
views
Which compact groups have finitely many irreducible representations of each dimension?
If my understanding is correct, this is true of sufficiently nice nonabelian Lie groups (see Ben Webster's answer below), and any finite group. On the other hand, this is false for any infinite ...
- 118k
12
votes
2
answers
883
views
Does almost every pair of elements in a compact Lie group generates the connected component?
It is known that almost every pair of elements in a connected compact Lie group (topologically) generates the group.
Obviously this isn't true for non-connected groups but
Given a compact Lie ...
- 262
9
votes
1
answer
657
views
Must an inverse limit of simply connected groups be simply connected?
While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...
- 7,193
9
votes
2
answers
717
views
Topology on extensions of topological groups
Let $G$ and $H$ be two topological groups and let $\mathcal{E}:0 \to G \to E \to H \to 0$ be an extension of abstract groups.
Is there a way to introduce a topology on $E$ such that $\mathcal{E}$ ...
- 125
9
votes
0
answers
367
views
Is every space a classifying space?
Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological ...
- 1,198
8
votes
1
answer
730
views
Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
- 1,414
8
votes
2
answers
362
views
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
- 41.9k
8
votes
2
answers
1k
views
$p$-adic exponentials for $p$-adic Lie groups
Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.
Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?
user113393
8
votes
0
answers
819
views
Second homotopy group of a topological group
It is well-known that any Lie group $G$ has $\pi_2(G)=0$: see this question. Is the same true for any compact (Hausdorff) topological group? Or even for locally compact ones? Maybe there is a way of ...
- 3,146
7
votes
4
answers
2k
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.] ...
7
votes
1
answer
490
views
Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...
- 13.6k
7
votes
2
answers
2k
views
Conditions for a topological group to be a Lie group
In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157):
Let $G$ be a locally compact group....
- 1,105
7
votes
1
answer
342
views
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 ...
- 863
7
votes
1
answer
270
views
Is $\Gamma(p) := \text{Ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{F}_p)$ a "standard" subgroup?
Let $\Gamma(p) := \text{ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{Z}_p/p))$.
Viewing $SL_2(\mathbb{Z}_p)$ as an analytic group, is there a formal group law $F$ in three variables, defined over $...
- 7,527
7
votes
1
answer
205
views
Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$
Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...
6
votes
2
answers
392
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 $G$...
- 11.3k
6
votes
2
answers
379
views
About Lie group $G$ has this escape property？
Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
...
- 71
6
votes
1
answer
1k
views
Classification of compact connected abelian groups
It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
- 1,586
6
votes
3
answers
1k
views
$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
- 17.5k
6
votes
1
answer
235
views
Hausdorff distance in compact Lie groups
Let $G$ be a compact Lie group with a compatible biinvariant metric $d$. The hyperspace $K(G)$ of nonempty compact subsets of $G$ is a compact metric space with the Hausdorff metric, and it is easy to ...
- 157
6
votes
1
answer
287
views
Are nearby crossed homomorphisms from compact Lie groups crossed-conjugate?
Charles Rezk had highlighted in MO:q/123624 that "Nearby homomorphisms from compact Lie groups are conjugate", and in consequence -- further highlighted in Remark 2.2.1 of his Global ...
- 19.8k
6
votes
1
answer
249
views
Growth function of locally compact groups
Every locally compact second countable group $G$ has a regular left-invariant measure $h$, the Haar measure. On the other hand the Birkhoff–Kakutani Theorem asserts that such groups also admit a ...
6
votes
0
answers
244
views
What can lattices tell us about lattices?
A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
- 2,183
6
votes
0
answers
341
views
When exponential map is 1-1 from vector fields to diffeomorphisms
Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
- 5,405
6
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?
- 4,351
5
votes
3
answers
1k
views
On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
- 7,609
5
votes
1
answer
356
views
Density of matrix coefficients of unitary representations of a locally compact group
Let $G$ be a locally compact group, $C_0(G)$ the $C^*$-algebra of continuous functions on $G$ that vanish at infinity, $C_b(G)$ the $C^*$-algebra of bounded continuous functions on $G$. We know that $...
- 1,586
5
votes
0
answers
184
views
Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$
Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$?
Apparently, there is an almost complete classification in ...
- 6,962
5
votes
0
answers
300
views
Matrix groups with two generators
Given two matrices $A,B\in{\rm{SL}}_2(\Bbb{R})$, is there any criterion guaranteeing that the subgroup they generate is discrete? What if one puts restrictions on $A,B$ e.g. they are both elliptic? ...
- 3,599
5
votes
0
answers
129
views
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 ...
- 29.1k
5
votes
0
answers
119
views
Characterizing Herz-Schur multipliers using coefficient functions of uniformly bounded representations
Let $G$ be a group and let $c > 1$ be a constant. We denote by $B_c(G)$ the space of all coefficients of the representations of $G$ which are uniformly bounded by $c$; more precisely, a function $f:...
- 299
4
votes
3
answers
2k
views
Why is the unipotent radical of a parabolic subgroup unimodular?
Let $G$ be a connected, reductive group defined over a local field $F$. Let $P$ be a parabolic subgroup of $G$ which is defined over $F$, and let $N = \mathscr R_u(P)$ be the unipotent radical of $P$....
- 6,180
4
votes
1
answer
344
views
Extensions of compact Lie groups
Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...
- 231
4
votes
2
answers
2k
views
A question about unitary and anti-unitary matrices
The question is the following: Let $U:\mathbf{C}^n\to \mathbf{C}^n$ be a unitary operator; let $\tilde{U}:\mathbf{C}^n\to\mathbf{C}^n$ be an antiunitary operator.
Can one deform $U$ to $\tilde{U}$ ...
- 327
4
votes
1
answer
304
views
Finite covolume of uniform lattice in quotient group
Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact.
Suppose ...
user215380
4
votes
1
answer
384
views
Invariant integration on principal bundles
Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \...
- 143
4
votes
0
answers
132
views
Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$
$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...
- 161
4
votes
0
answers
640
views
Closed subgroups of a connected Lie group
Is it true that for any closed subgroup $H$ of a connected Lie group $G$, the group of connected components $\pi_0(H)$ is finite or countable? (inspired by the comment of nfdc23 to this question ).
- 14.2k
4
votes
0
answers
264
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 ...
- 6,287
3
votes
1
answer
197
views
Is the exponential map of a locally compact group a local homeomorphism?
We consider a locally compact abelian group $G$. We equip the real vector space $A(G)$ of continuous group homomorphisms $\mathbb{R}\to G$ with the topology of uniform convergence on compact subsets ...
- 3,031
3
votes
1
answer
454
views
Principal bundles from a fibration of homogeneous spaces
Let $G$ be a compact (Lie) group, and $H \subseteq H'$ two compact (Lie) subgroups. It is clear that we have an obvious surjective map of homogeneous spaces
$$
G/H \twoheadrightarrow G/H'.
$$
Will it ...
3
votes
1
answer
201
views
Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally compact Abelian groups?
It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$.
Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or ...
- 41.9k
3
votes
1
answer
122
views
A converse of Cartan's automatic continuity theorem
Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...
- 35
3
votes
1
answer
380
views
A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$
Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...
- 356
3
votes
1
answer
141
views
Measure on orbits of $N$ under conjugation by $H$
Let $G$ be a locally compact topological group with closed subgroups $H, N$ and $H$ normalizing $N$. Then $H$ acts continuously on $N$ by conjugation. If it will help, assume that $N$ is nilpotent, ...
- 6,180
3
votes
1
answer
142
views
What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite
When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
- 1,040
3
votes
0
answers
58
views
Criteria for density of subgroup of diffeomorphism group
Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
- 5,405
3
votes
0
answers
335
views
Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions
Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...
- 3,652