5
votes
1answer
185 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
1answer
193 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
1answer
414 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
0answers
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
3answers
334 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
0answers
223 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
1answer
192 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
1answer
154 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
1answer
100 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
3answers
291 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
2answers
203 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
4answers
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
2answers
175 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
1answer
174 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
1answer
112 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
1answer
256 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
1answer
205 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
2answers
461 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
2answers
738 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
1answer
470 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
0answers
176 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) ...
5
votes
4answers
515 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 ...
16
votes
2answers
924 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
1answer
450 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
2answers
948 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 ...
4
votes
2answers
1k 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
1answer
586 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
1answer
442 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
2answers
712 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
4answers
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 ...
13
votes
6answers
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
0answers
241 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 ...
3
votes
1answer
687 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
1answer
542 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
1answer
354 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
2answers
873 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
4answers
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
1answer
139 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
1answer
287 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
2answers
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
3answers
757 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). ...