# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $H$ ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...