# Tagged Questions

**5**

votes

**1**answer

211 views

### Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...

**5**

votes

**1**answer

207 views

### Determining the Lie algebra elements exponentiating to the center of a Lie group

For a semi-simple compact Lie group $G$ with center $Z(G)$, one can characterize the preimage of $Z(G)$ in the Cartan subalgebra under the exponential map as the nodes of the Stiefel diagram (see for ...

**1**

vote

**1**answer

424 views

### Para-Complexification of Lie Groups

Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...

**0**

votes

**0**answers

55 views

### compute the determinant of a conjugacy map

Let $k$ be an algebraically closed field.
Let $F=k((\pi))$ and $\mathcal{O}$ the ring of integers, let $\gamma\in T(F)$ regular semisimple for a connected reductive group $G$.
We consider the map ...

**9**

votes

**3**answers

353 views

### Diffeomorphisms of the sphere conjugate to a rotation

What are sufficient condition on a given diffeomorphism of the sphere (say, given explicitly with formulas) that can ensure that it is conjugate to a rotation, in the group of diffeomorphism of the ...

**5**

votes

**0**answers

231 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 ...

**2**

votes

**1**answer

193 views

### What is the name of this product of Lie groups

Let $G$ be a Lie group and $V$ be a vector space. Let $\rho_{l} : G \times V \to V$ be a left representation and $\rho_r:V \times G \to V$ be a right representation which commutes with $\rho_l$ in the ...

**3**

votes

**1**answer

163 views

### Visualizing Bianchi type/homogenous spaces

I'm aware of Bianchi's (local) classification of homogenous 3-manifolds into the Bianchi types I through IX, and I can follow the algebra for classifying the Lie algebras. However, I still can't ...

**3**

votes

**1**answer

104 views

### Symmetry group for the frame bundle of a G-space

Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left.
Question 1: does the group $G$ act naturally on the tangent bundle $TQ \to Q$?
My motivation here ...

**3**

votes

**3**answers

293 views

### What are the symmetries of a principal homogeneous bundle?

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:
$G$ is a Lie group, and $H$ ...

**5**

votes

**2**answers

213 views

### Submersions from compact flat manifold

Let $M=\mathbb{R}^n/G$ be a closed flat manifold, and let $F\to M \to N$ be a locally trivial submersion, where $F$ and $N$ are closed manifolds.
My question is simple: are $F$ and $N$ homeomorphic ...

**27**

votes

**4**answers

2k views

### What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...

**1**

vote

**2**answers

183 views

### Copies of ax+b inside the AN part of an Iwasawa decomposition?

As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...

**1**

vote

**1**answer

176 views

### Non-trivial action of $SL_n(\mathbb{Z} )$ on a simplicial tree

A group $G$ has Serre's property $FA$ if any isometric action of $G$ on a simplicial tree has a global fixed point. Let $n\geq 3$. It is well-known that $SL_n(\mathbb{Z} )$ has property $FA$. Now my ...

**1**

vote

**1**answer

115 views

### How to judge a manifold generated by Coxeter system smooth?

For the definition of Coxeter System, you can see: http://en.wikipedia.org/wiki/Coxeter_group
Given a chamber Q, given a Coxeter System $(\Gamma,V)$, we can defined a set M by the following way: ...

**1**

vote

**1**answer

272 views

### Extension of groups in Bieberbach's theorem

I am reading de la Harpe's book "Topics in Geometric Group Theory".
On page 145, there is a theorem:
Let $V$ be a complete $n$-Riemannian manifold with sectional curvature satisfying $K\ge 0$. Then ...

**2**

votes

**1**answer

210 views

### How many quotients can a finitely generated group have or how many bundles over aspherical spaces does a fixed total space support?

Consider $M^3_{pq}$,
a torus bundle over $S^1$ with fundamental group the HNN extension generated by three generators $x,y,z$ satisfying the relations $\quad [x,y], \quad x^z = x^p \quad$ and $y^z = ...

**6**

votes

**2**answers

471 views

### cohomological dimension of a group acting on a product

I recently came across an interesting result of Kobayashi [Corollary 5.5], a special case of which is the following: Suppose $\Gamma$ is a discrete torsion free subgroup of $SL_n(\mathbb{R})$ which ...

**13**

votes

**2**answers

795 views

### Regarding Cayley Graphs of Property (T) Groups

A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...

**4**

votes

**1**answer

491 views

### Random geometries

Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all ...

**3**

votes

**0**answers

181 views

### Equivariant Tangent Bundle Decomposition

Given a $G$-homogeneous space $M$, for $G$ a (Lie) group, we have a canonical $G$-action on the tangent bundle $T(M)$ of $M$. If $M$ is a complex manifold, then we have a decomposition of $T(M) ...

**6**

votes

**4**answers

529 views

### On the determination of a quadratic form from its isotropy group

Let $F:\mathbf{R}^n\rightarrow\mathbf{R}$ be a non-degenerate quadratic forms. Let
$$
O(F):=\{g\in GL_n(\mathbf{R}):F(gv)=F(v),\forall v\in \mathbf{R}^n\}
$$
be the isotropy group of $F$.
Q: So how ...

**17**

votes

**2**answers

961 views

### Can every Lie group be realized as the full isometry group of a Riemannian manifold?

Suppose a finite--dimensional Lie group $G$ is given. Does there exist a manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$?
For example if I try to do this ...

**11**

votes

**1**answer

462 views

### Is $SL(n,\mathbb{Z})$ a CAT(0) group?

Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.

**5**

votes

**2**answers

1k views

### one-parameter subgroup and geodesics on Lie group

Hi,
Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...

**5**

votes

**2**answers

2k views

### Why SU(3) is not equal to SO(5)?

I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are ...

**2**

votes

**1**answer

632 views

### Taylor's series for Lie groups

Let $G_1$ and $G_2$ be two (matrix) Lie groups, with $L(G_1)$ and $L(G_2)$ their respective Lie algebras.
I am interested to know if there is a well developed theory to approximate a (sufficiently) ...

**3**

votes

**1**answer

456 views

### Uniform lattices in semisimple Lie groups

Let $\Gamma$ be a uniform lattice in a semisimple Lie group $G$.
Must $\Gamma$ be virtually torsion-free?
If (1) is true, then does this work more generally if $G$ is reductive?
I am motivated by ...

**7**

votes

**2**answers

752 views

### Affine manifolds

An affine manifold is a topological manifold which admits a system of charts such that the coordinate changes are (restrictions of) affine transformations. Let $M$ be a compact affine manifold. Let ...

**5**

votes

**4**answers

1k views

### Lie groups admitting flat (bi)invariant metrics.

I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric.
Is there any non-abelian compact lie group admitting a flat metric that is bi or ...

**15**

votes

**7**answers

3k views

### Topology of SU(3)

$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...

**3**

votes

**0**answers

252 views

### Higher order Pansu derivative

Given a group $(G,*)$ there is no candidate for what can be understood as a derivative of a function $$f:G\rightarrow\mathbb{R}.$$ However, for the special case of Carnot groups there is the ...

**4**

votes

**1**answer

735 views

### Classification of discrete subgroups of the unitary group

Let $U(n)$ be the unitary group. From André Weil's paper "On discrete subgroups of Lie groups" it is well known that discrete cocompact subgroups of $U(n)$ have only a finite number of generators and ...

**0**

votes

**1**answer

568 views

### Action of $SL(2,\mathbb{C})$ on representations of $SU(2)$

I want to precisely understand in what sense is (if it is!) $SL(2,\mathbb{C})$ the "complexified" version of $SU(2)$?
Can I think of it like choosing a natural matrix basis of the real three ...

**12**

votes

**1**answer

362 views

### Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...

**9**

votes

**2**answers

933 views

### Existence of finite index torsion free subgroups of hyperbolic groups

Question. Is it true that each infinite hyperbolic group
has a torsion free subgroup of finite index?
Are there counterexamples, or positive results for some large subclasses of hyperbolic groups?
...

**5**

votes

**4**answers

1k views

### Which Riemannian manifolds admit a finite dimensional transitive Lie group action?

This is a basically an adjusted version of my earlier question about how to define a convolution algebra on a general Riemannian manifold. The motivation for asking such a question of course comes ...

**1**

vote

**1**answer

141 views

### A question about iterated quotients in riemannian geometry

Background
This can be generalised, but let me be fairly concrete. Let $X$ be a simply-connected riemannian manifold and let $G$ denote the Lie group of isometries, assumed nontrivial. Let $F < ...

**0**

votes

**1**answer

296 views

### What kind group can be realized as a Isometry group of some space?

Every group G is a subgroup of Isometry group of its Cayley graph.
What is essential property of being an Isometry group?
Lie group?

**4**

votes

**2**answers

1k views

### Poincaré Theorem on presentation from a fundamental polyhedra

Poincaré Theorem on Kleinian groups (groups acting discontinously on Euclidean or hyperbolic spaces or on spheres) provides a method to obtain a presentation of a Kleinian group from a fundamental ...

**10**

votes

**3**answers

819 views

### Non-Lie Subgroups

A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof).
...