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?