I want to do an Erlangen approach to classical Geometry and below I discuss how $PO(n+1,1)$ acts transitively on a model for hyperbolic space. I want help finding the stabilizer under this actions and to calculate the quotient group, I wan to check that this quotient group is isomorphic to $SO^+(2,1)/SO(2)$, coming from the Lorents group of signature $(2,1)$ acting on the hyperboloid model.
I define the quadratic form $q=\sum_{i=1}^{n+1}x_i^{2}-x_{n+2}^2$ and I denote by $O(n+1,1)$ the orthogonal group w.r.t the polar form associated with $q$. I denote by $PO(n+1,1)$ the induced group on my projective space $P^{n+1}( \mathbb{R})$ and I let $C= \bar{q}^{-1}(0)$ where $\bar{q}$ is the projective image of $q$. I do the following identification $C\rightarrow S^n$, $[\frac{x_1}{x_{n+2}}, \frac{x_2}{x_{n+2}},\dots \frac{x_{n+1}}{x_{n+2}},1]\rightarrow (\frac{x_1}{x_{n+2}}, \frac{x_2}{x_{n+2}},\dots \frac{x_{n+1}}{x_{n+2}})$ (note that $x_{n+2}\neq 0$).
The projective transformations that fixes $C=S^n$ in $P^{n+1}(\mathbb{R})$ are precisely $PO(n+1,1)$, which acts transitevely on $S^n$ as well as the open disc $D^n$, in particular the open disc is a model of hyperbolic $n+1$ space (Klein model). The stabilizer of a point in $D^n$ under this action can be realized to be the affine transformations fixing $S^n$. I.e fixing an interior point is equivalent to fixing the polar hyperplane of the point which does not intersect $S^n$. Hence as sets there is a bijection $H^{n+1}=(PO(n+1,1)=O(n+1,1)/\pm I)/O(n+1)$
In the Hyperboloid model of the hyperbolic plane we consider Minkowski space of signature $(2,1)$ and the negative radius unit sphere, this is a two sheeted Hyperboloid and our model space is the upper sheet. Hence we consider the subgroup of $O(2,1)$ preserving this sheet, as I understand it is denoted $SO^+(2,1)$, the stabilizer, preserving a point on the hyperbola (time-like) vector, has two space like dimensions to move in and should be SO(2).
My question becomes then the following is $SO^+(2,1)/SO(2)\cong (O(2,1)/\pm I)/O(2)$?
Or is there a mistake somewhere in the line of thought?
Please help me out!