Background
On page 232 of [1], the authors mentioned a few examples of Fuchsian groups that could be explicitly written down, namely, four cyclic groups; Hyperbolic, Parabolic, Elliptic, and Modular. Yet, they haven't put concrete examples of each so I started working on myself and I got stuck at understanding hyperbolic cyclic groups.
Question
From the wiki, I understood the definition of hyperbolic cyclic groups link. And it seems the Dihedral group of order 8 (as it's a finitely generated group of Mobius transformations, rotations and reflections) meet the definition.
But I was not sure why it's called Hyperbolic... could someone explain it to me?
Reference
[1] Jones, Gareth A., and David Singerman. Complex functions: an algebraic and geometric viewpoint. Cambridge university press, 1987.