0
votes
1answer
147 views

Compact Lie groups with only 3 dimensional cohomology generators

Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd. For which $M$, ...
1
vote
1answer
112 views

Structures on open surfaces

Let $\phi\in PSL(2,R)$ be hyperbolic and $\varphi\in PSL(2,R)$ be elliptic. Is it possible to find a local homeomorphism $f:H^2\rightarrow H^2$ such that $f(\phi(x))=\varphi(f(x))$ for all $x\in H^2$ ...
1
vote
1answer
135 views

Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure (In sense of ...
6
votes
0answers
120 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...
6
votes
1answer
203 views

Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact: An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
0
votes
1answer
181 views

Coadjoint orbits and homogeneous symplectic $G$-manifolds

We know this important fact from A.A.Kirillov that : Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...
5
votes
0answers
229 views

Quotient of 3-sphere by binary octahedral group?

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...
4
votes
1answer
380 views

Transitive action on the sphere

Hello, One of the subgrouops of $SO(n)$ which acts transitively on the sphere $S^{n-1}$ is the (compact) symplectic group $Sp(n/4)$. The center of $Sp(m)$ is isomorphic to $\mathbb{Z}_2$. Can we embed ...
1
vote
1answer
160 views

Orbits of Product Lie Groups Action

Hi to all, Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
10
votes
1answer
211 views

orbit space of a topological manifold

Given a compact Lie group G acting freely on a topological manifold M, is it true that the orbit space M/G is also a topological manifold? If so, why?
4
votes
2answers
268 views

Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
5
votes
3answers
392 views

Matrices generating non-discrete subgroups of SL(2,R)

Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
22
votes
2answers
1k views

Simple discrete subgroups of Lie groups

Upon Ian Agol's suggestion, I separated this question from the one on non-residual finiteness in Non-residually finite matrix groups Question. Are there infinitely generated simple discrete ...
5
votes
1answer
138 views

Algorithmic Borel Finiteness?

It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
0
votes
0answers
109 views

Generating Set for Og over $\mathbb Z_2$

Hi, All: I am reading a claim that Og , the orthogonal group associated with a finite-dimensional vector space V over $\mathbb Z_2$ , and a quadratic form q defined therein , i.e., the group of ...
5
votes
2answers
525 views

Finite-dimensional subgroups of diffeomorphism groups

This question is a generalization of my previous question about the circle to arbitrary manifolds. Is there a smooth manifold M with the following property. There exists a sequence of connected ...
7
votes
2answers
700 views

Finite-dimensional subgroups of circle diffeomorphism group

Is there a sequence of connected finite-dimensional subgroups Gi of the circle diffeomorphism group G with the following properities: (a) Gi is contained in Gj for i < j (b) The union of Gi is ...
5
votes
1answer
511 views

Which groups admit a unique Lie group structure?

This question is a follow-up on the answer given here Can a Lie group as an abstract group be given more than one topology making it a Lie group? It is motivated by the following observations: If ...
1
vote
2answers
312 views

module of sections of the horizontal bundle

Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...
2
votes
2answers
359 views

Is every group object in TopMan a Lie group?

Recall that a Lie group is a group object in the category of C∞ manifolds. If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure ...