Isometries of Hyperbolic Space

Thinking of 2-dimensional hyperbolic space in the Poincare Disk Model, (nonidentity, orientation preserving) isometries fall into three categories: hyperbolic, parabolic, or elliptic - depending on the number of fixed points and whether the fixed point(s) lie on the boundary of H^2 or in the space itself. Our goal here is to visualize the space of all such isometries, and understand how each of these three categories live in that space.

Hyperbolic Isometries

Hyperbolic isometries have exactly 2 fixed points both on the boundary of H^2. To completely determine one such isometry, one needs three piece of information: repelling fixed boundary point, attracting fixed boundary point, the translation amount along the geodesic joining these points. So topologically, this is a subset of S^1 x S^1 x R+. As the translation amount gets closer to 0, all the isometries get closer to the identity. Moreover, the boundary points must be distinct, so the diagonal of S^1 x S^1 is missing.

Another detail to notice is any two hyperbolic isometries with the same translation amount are conjugate to each other (conjugate by the appropriate rotation about the origin). Topologically, one such conjugacy class is a torus minus the diagonal. Cutting open the torus along the diagonal yields a cylinder. After some thought, we really should cut open along the diagonal and stretch the space into a cylinder, since isometries on opposite sides of the diagonal should not be "near" each other at all.

Parabolic Isometries

Parabolic isometries have exactly 1 fixed point on the boundary of H^2. To completely determine one such isometry, one needs two pieces of information: fixed boundary point, and amount of translation along horocycles tangent to that point. The latter can be any nonzero real number (some care must be taken to make this a well defined value for any parabolic). As the translation amount gets close to zero, the isometries get closer to the identity. Note that any two parabolics are conjugate to each other.

Elliptics Isometries

Elliptic isometries have a single fixed point in the space H^2 itself. These are determined by the fixed point and the amount of rotation around this fixed point (again, some care must be taken to make this rotation value well defined). As the rotation amount nears 0 or 2pi, the isometry gets close to the identity. Hence we have this interior of a croissant shapes worth of isometries. Elliptics are conjugate if they have the same rotation amount (conjugate by a hyperbolic taking old fixed point to new fixed point). Hence a conjugacy class here looks topologically like an open disk.

How do these glue together?

In order for a path of hyperbolic isometries to converge to a parabolic, one needs the distance between the fixed points to be getting smaller. There are two ways this could happen, depending on the orientation of the attracting vs. repelling fixed points. However, if the translation amount of the hyperbolic isometries in the path is not also going to zero, the isometries aren't converging to anything. One needs both the translation amount along the fixed geodesic AND the distance between the fixed points to be going to zero in order to converge to a parabolic isometry. The rate at which these two are converging to zero determines the amount of translation in the resulting parabolic. Hence a rainbow shaped cylinder drawn here is NOT a conjugacy class, but a set of hyperbolics converging to a set of parabolics with + or - the same translation amount. The boundary of a hyperbolic conjugacy class is on the boundary of the entire space.

There is a similar story for how the elliptics glue onto the parabolics. A path of elliptic isometries only converges to a parabolic only if the interior fixed point is converging to a point on the boundary, AND the amount of rotation is converging to zero. The rate at which this happens determines the translation amount of the resulting parabolic. Since we've drawn the hyperbolics glueing onto the outside of the parabolic cones, we can visualize the elliptics glueing onto the inside. There is a similar note of caution about the conjugacy classes here. A conjugacy class of elliptics has boundary on the boundary of the entire space, as well.

We can redraw the picture on the right as the interior of a torus!

purple / pink = hyperbolic conjugacy classes

blue = parabolics

green / yellow = elliptic conjugacy classes

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