Volume 14, issue 3 (2010)

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Affine deformations of a three-holed sphere

Virginie Charette, Todd A Drumm and William M Goldman

Geometry & Topology 14 (2010) 1355–1382

Associated to every complete affine 3–manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface Σ. We classify these complete affine structures when Σ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Σ, the deformation space identifies with two opposite octants in 3. Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4, ).

hyperbolic surface, affine manifold, discrete group, fundamental polygon, fundamental polyhedron, proper action, Lorentz metric, Fricke space
Mathematical Subject Classification 2000
Primary: 57M05
Secondary: 20H10, 30F60
Received: 7 October 2009
Revised: 10 May 2010
Accepted: 23 April 2010
Published: 4 June 2010
Proposed: Walter Neumann
Seconded: Jean-Pierre Otal, Benson Farb
Virginie Charette
Département de mathématiques
Université de Sherbrooke
Québec J1K 2R1
Todd A Drumm
Department of Mathematics
Howard University
Washington DC 20059
William M Goldman
Department of Mathematics
University of Maryland
College Park MD 20742