Kleinian groups: From limit sets to deformation spaces

The first volume of Fricke-Klein Vorlesungen über der Theorie der automorphen Funktionen contains some remarkable drawings of the limit sets (Grenzegebilde) of groups generated by reflections in various configurations of tangent circles.

We begin by reviewing the construction of these pictures from a modern viewpoint. A particular feature is the presence of many round circles in the limit sets.

We will discuss the meaning of these circles in terms of three dimensional hyperbolic geometry and how this led to a method of locating the fractal boundary of the quasi-conformal deformation spaces of various families of Kleinian groups.