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.
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.