TITLE: Why are Fuchsian groups interesting?
QUESTION [28 upvotes]: I keep hearing that fuchsian groups are interesting for other reasons than the Fuchsian model for hyperbolic Riemann surfaces. 
What are those reasons? 
Are the Fuchsian groups with fixed points interesting from a geometric perspective?
Where do Fuchsian groups appear besides hyperbolic geometry? 
I also read somewhere about a relation between fuchsian groups and fractals. Does someone know more about that and/or has a good reference?

REPLY [7 votes]: Are the Fuchsian groups with fixed points interesting from a geometric perspective?

Yes, notice $PSL(2,\mathbb{Z})$ fixes points. The fundamental domain in this case is isometric to a hyperbolic structure on the disk with two cone points. 
More generally, if a Fuchsian group $G$ has finite co-volume in $H^2$, then $H^2/G$ will be a hyperbolic 2-orbifold. For example, all of the orientation preserving subgroups of the hyperbolic triangle groups are Fuchsian groups of this type.
In particular, hyperbolic triangle groups have a unique hyperbolic structure, so if 3-manifold has quotient which contains a (orientable) hyperbolic triangle orbifold then the original 3-manifold contains a totally geodesic surface.