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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

Another good explanation with examples is available in this article section 12.