The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

Abstract

For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a ``hyperelliptic'' three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the {\sl Weierstrass lines} of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six {\sl generalized Weierstrass points} on the surface.