The Geometry of Two Generator Groups: Hyperelliptic Handlebodies
 Jane Gilman,
 Linda Keen
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A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. 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 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 generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.
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 Title
 The Geometry of Two Generator Groups: Hyperelliptic Handlebodies
 Journal

Geometriae Dedicata
Volume 110, Issue 1 , pp 159190
 Cover Date
 20050201
 DOI
 10.1007/s1071100465568
 Print ISSN
 00465755
 Online ISSN
 15729168
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Fuchsian groups
 Kleinian groups
 Schottky groups
 Riemann surfaces
 hyper elliptic surfaces
 Industry Sectors
 Authors

 Jane Gilman ^{(1)}
 Linda Keen ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Rutgers University, Smith Hall, Newark, NJ, 07102, U.S.A.
 2. Mathematics Department, CUNY Lehman College and Graduate Center, Bronx, NY, 10468, U.S.A.