Lie groups for beginners: Lie group of hyperbolic geometry

Question

I am trying to understand Lie groups and their relation to (2 dimensional) hyperbolic geometry.

as far as I understand it (which is not very far, I am pushing my understanding here) the Lie-group is the set of all isometric transformations in a geometry.

So in hyperbolic geometry this is the set of all reflections , translations, rotations, horolation and maybe other (hyperbolic) length preserving transformations.

But then

What does it mean that "the transformation group of hyperbolic geometry is the Orthochronous Lorentz group $O ( 1 , n ) / O ( 1 ) $ ?" (found for example at https://en.wikipedia.org/wiki/Klein_geometry#Examples )

As far as I can follow it (and I am pushing my understanding here) it should depend on which mode of hyperbolic geometry you use.

In the Poincare disk model the transformation group is the set of 1) all circle inversions in circles orthogonal to the boundary circle and 2) their combinations. (the first one being reflections in hyperbolic lines, the second one multiple reflections.

In the Poincare half plane model they are another transformation group. the set of 1) all circle inversions in circles centered on the boundary circle 2) reflections in lines orthogonal to the boundary line and 3) their combinations. (the first two being reflections in hyperbolic lines, the third one multiple reflections.)

But then I got stumped what does this has to do with the lorentz group? or any other named group $ SO(2)$ , $ SO(2)$or $SO^+(2)$ or $ O ( 1 , n ) / ( O ( 1 ) × O ( n ) ) $ (from https://en.wikipedia.org/wiki/Hyperbolic_geometry#Homogeneous_structure , I guess n= 2 here but I don't even understand the formula)?

I could do with a basic "Introduction to Lie groups for hyperbolic critters" book, recommendations welcome.

Answer 1

There are many ways to think about hyperbolic geometry, but for me, the hyperbolic plane is a two dimensional complete simply-connected Riemannian manifold $(M,g)$ with constant curvature $-1$.

Given this definition, one can ask two questions. The first is whether such a space exists and the second is whether those properties determine the space uniquely up to an isometry in the sense that if $(M_1,g_1)$ and $(M_2,g_2)$ are two dimensional complete simply-connected Riemannnian manifolds with constant curvature $-1$ then $(M_1,g_1)$ and $(M_2,g_2)$ are isometric.

Treating the existence, one can construct such a space in various ways (which are called models of the hyperbolic plane). The model relevant to your question is called the Hyperboloid model for which $M$ is taken to be forward pointing sheet of a hyperboloid sitting inside the Minkowski space $\mathbb{R}^{1+2}$ and the metric $g$ is the Riemannian metric on $M$ induced on $M$ from the pseudo-Rimennaian Minkowski metric on $\mathbb{R}^{1+2}$.

Since $M$ sits inside $\mathbb{R}^{1+2}$, any isometry $\varphi$ of $\mathbb{R}^{1+2}$ that fixes $M$ (satisfies $\varphi(M) = M$) will descend to an isometry of $M$. The group of linear isometries of $\mathbb{R}^{1+2}$ is a subgroup of $GL_3(\mathbb{R})$ called the Lorentz group and is denoted by $O(1,2)$. The subgroup of $O(1,2)$ fixing $M$ is called the orthochronous Lorentz group and is denoted by $O^{+}(1,2)$. It turns out that $O^{+}(1,2)$ is the full group of isometries of $M$ and $O^{+}(1,2)$ is isomorphic to $O(1,2)/O(1)$, hence the description you ask about.

Besides the hyperboloid model, one can construct many other explicit models for the hyperbolic plane, show that they satisfy the definition given in the beginning and show explicitly that each pair of models are indeed isometric. In particular, this implies that the isometry groups of different models should be isomorphic. For example, in the Poincaré upper half-plane model, orientation preserving isometries are interpreted as Möbius transformations and the group of orientation preserving isometries is naturally identified with $PSL(2,\mathbb{R})$. Since the hyperboloid model and the upper half-plane model are isometric, the groups of orientation preserving isometries in both models should be isomorphic and indeed $PSL(2,\mathbb{R}) \cong SO^{+}(1,2)$. Thus, when you read about different models for the hyperbolic plane, you may find different descriptions of the isometry groups but they all will be isomorphic.

Finally, a non-trivial result states that indeed the definition stated in the beginning determines the space uniquely up to an isometry so it's not a coincidence that all the models one usually deals with for the hyperbolic plane are isometric.