I'm not entirely sure what I'm trying to ask.
According to my understanding of the Erlangen programme, each "geometry" (in the sense of Euclidean or hyperbolic or elliptic geometry) is defined in some sense by its abstract group of congruences. My question is whether there's a way to go from the abstract group to the "salient" objects of the geometry.
Examples, because I'm not sure what I mean:
The "salient" objects in Euclidean plane geometry are the points, lines and circles (I suppose). The lines and points correspond in some sense to the reflections, because the set of fixed points of a reflection is either a point or a line. While circles aren't fixed points of any Euclidean congruence transformation, they are the result of rotating a point around.
The Euclidean example suggests we can get the "salient" objects either via fixed points or via orbits of subgroups.
In the case of the elliptic plane, I know that the isometry group is $SO(3)$. When a transformation in $SO(3)$ acts on $\mathbb R^3$, its set of fixed points is a line through the origin. These are the points of the elliptic plane. But elliptic geometry has also got lines and circles. Circles are the result of spinning a point around.
Question: Is there a systematic way of deriving the "salient" objects from an abstract group?
Best Answer
If $G$ is a reductive Lie group, then it may be viewed as a group of symmetries for a geometry whose salient objects are the coset spaces $G/P$ with respect to parabolic subgroups $P$. This point of view is alluded to in TWF249, worked out in detail for $G = \operatorname{PGL}(3)$ (as mentioned in TWF250, not the isogenous group $\operatorname{SL}(3)$, despite what's claimed in TWF178) in TWF178, and sketched for some more exotic groups in TWF187.
In TWF249, Baez refers to double coset spaces, but my understanding of the surrounding discussion seems actually to emphasise single coset spaces (which are anyway nicer, because $G$ acts on them!). Anyway, since the question is open ended, perhaps an open ended answer is also acceptable.