# classifications of isometries of $\mathbb{H}^2$

Let $\mathbb{H}^2$ be the hyperbolic plane and let $\phi \mathbb{H}^2$ be the boundary at infinity of $\mathbb{H}^2$. Let the union $\mathbb{H}^2 \cup \phi \mathbb{H}^2$ be donoted by $\alpha \mathbb{H}^2$. If $f \in Isom(\mathbb{H}^2)$, and $f$ can uniquely be extended to a map $F : \alpha \mathbb{H}^2 \rightarrow \alpha \mathbb{H}^2$, then we can classify the following:

Elliptic: If $F$ fixes a point $p \in \mathbb{H}^2$ then $f$ is called elliptic.

Parabolic: If $F$ has exactly one fixed point in $\phi \mathbb{H}^2$ then $f$ is parabolic.

Hyperbolic: If $F$ has two fixed points in $\phi \mathbb{H}^2$, then $f$ is hyperbolic.

Identity: If $F$ has at least three fixed points in $\alpha \mathbb{H}^2$, then $f$ is the identity.

Note that "isom" is isometry. I guess my question is that I can not picture or understand the cases that occur, especially the last one. I was hoping someone could explain it to me to help me understand what is going on better.

• By "last one", you mean the identity? That's the map that doesn't change anything. The statement is that if it fixes three points on the boundary, it fixes everything on both the boundary and in $\mathbb{H}^2$ itself. – MartianInvader Jan 27 '14 at 23:13
• For the last, note that a point can be uniquely determined given its distance from three disticnt points. – AlexR Jan 27 '14 at 23:13
• Are you familiar with the upper-half-plane model of this in the complex plane? – Will Jagy Jan 27 '14 at 23:34
• @AlexR: That doesn't quite cut it, because one or more of the three given fixed points might be at infinity. – Henning Makholm Jan 27 '14 at 23:41
• @HenningMakholm You are surely correct, It was just meant as a hint for imagining why it could be. – AlexR Jan 28 '14 at 0:13

If you just want examples, a rotation about a point is elliptic; if you have a line $\ell$ and slide along $\ell$ from a point $P$ to a point $P'$, that’s hyperbolic (and the fixed points on the boundary are where $\ell$ hits it); it’s harder, I think, to give an abstract example of a parabolic, but in the plane model of $\mathbb H^2$, a rightward shift of everything by $1$ is an example. In the last case, the single point $\infty i$ on the boundary is the fixed point.
A nice explanation is given on pages $3$ and $6$ of Javier Aramayona's lecture notes Hyperbolic Structures on Surfaces which are part of the "Geometry, Topology And Dynamics Of Character Varieties" Lecture Notes Series.