Isometries of n-dimensional hyperbolic space

Question

I'm aware that the isometries of hyperbolic 2 dimensional and 3 dimensional space are given by elements of $SL(2,\mathbb{R})$ and $SL(2,\mathbb{C})$ respectively and are then categorised according to rules on the trace of the element, or equivalently the number of fixed points of the element. My question is whether there exists a form of isometries of n dimensional hyperbolic space and a similar way to make sense of this in terms of a function and number of fixed points (or planes etc.)?

Answer 1

In some sense the formulas $\text{Isom}(\mathbb{H}^2) \approx SL(2;\mathbb R)$ and $\text{Isom}(\mathbb{H}^3) \approx SL(2;\mathbb C)$ are exceptional.

There is a general formula for $\text{Isom}(\mathbb{H}^n)$ but it does not fit into these two patterns. Instead, $\text{Isom}(\mathbb H^n)$ is the Lorentz group $$\text{Isom}(\mathbb{H}^n) \approx SO(n,1;\mathbb R) $$ This notation refers to the group of $(n+1)$-by-$(n+1)$ real matrices $M$ such that $M^T J M = J$ where $J$ is the diagonal matrix with all $1$'s on the diagonal except for a single $-1$ on the lowest-right entry. What this says geometrically is that $\text{Isom}(\mathbb H^n)$ is the group of linear transformations of $\mathbb R^{n+1}$ with coordinate functions $(x_1,...,x_n,t)$ such that the Lorentz form $dx_1^2+...+dx_n^2-dt^2$ is preserved. This group action may be viewed as the isometries of the hyperboloid model $$\mathbb H^n = \{(x_1,...,x_n,t) \mid x_1^2+...+x_n^2-t^2=-1, \,\,\, t > 0\} $$ where the hyperbolic metric on $\mathbb H^n$ is given by restricting the Lorentz form to the tangent bundle $T \mathbb H^n$ (one may prove that the restriction produces a complete Riemannian metric of constant sectional curvature $-1$).

One can then derive linear algebra characterizations for various special types of isometries, in the same fashion as the characterization is obtained for $SL(2,\mathbb R)$ or $SL(2,\mathbb C)$. Namely, one classifies isometries by their fixed point sets, and one uses an eigenvalue analysis to convert those classification properties into matrix properties.