All Questions
26 questions
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
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
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
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
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
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
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
0
votes
0
answers
267
views
Definition of reducible lattice
I am reading Raghunathan's book on discrete subgroups of Lie groups.
In particular I am stuck on Corollary 5.19 which gives several equivalent conditions for a lattice in a semisimple Lie group to be ...
- 101
2
votes
1
answer
82
views
Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
- 25.8k
2
votes
1
answer
216
views
How to prove that Chevalley groups over $\mathbb R$ have no compact factors
I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem.
I've been told in another thread ...
- 332
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
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
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
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
2
votes
0
answers
91
views
Fixed point set with non-empty interior
Let $G$ be an infinite compact separable Hausdorff metric group, and $H\subset G$ a closed subgroup, such that the left $G$-action on $G/H$ is effective (i.e., $H$ doesn't contain a non-trivial closed ...
- 1,959
1
vote
0
answers
121
views
A section over an orbit space
Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...
- 1,959
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
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 ...
0
votes
1
answer
434
views
Reference request: Any connected Lie group has a countable base for its topology
I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to this question. Has any proof of this assertion been published?
- 14.2k
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
2
votes
0
answers
321
views
Surjective homomorphisms of non-connected Lie groups
Let $\psi\colon B\to C$
be a homomorphism of real Lie groups, where the group $C$ is connected.
Let $B^0$ denote the identity component of $B$, and we set $\pi_0(B)=B/B^0$, then $\pi_0(B)$ is a ...
- 14.2k
2
votes
0
answers
82
views
Uniquely divisible neighborhoods of identity in topological groups
Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a ...
- 1,959
2
votes
0
answers
62
views
Codistal subgroups of locally compact groups
Let $G$ be a topological group and let $H$ be a closed subgroup of $G$. Say $H$ is codistal in $G$ if the translation action of $G$ on the coset space $G/H$ is distal (meaning that no non-diagonal ...
- 4,728
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
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
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