# Tagged Questions

**0**

votes

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**3**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**2**answers

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 ...