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
Abstract

Associated to every complete affine 3–manifold $M$ with nonsolvable fundamental group is a noncompact hyperbolic surface $\Sigma$. We classify these complete affine structures when $\Sigma$ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface $\Sigma$, the deformation space identifies with two opposite octants in ${ℝ}^{3}$. 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 $Sp\left(4,ℤ\right)$.

Keywords
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
Publication
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
Authors
 Virginie Charette Département de mathématiques Université de Sherbrooke Sherbrooke Québec J1K 2R1 Canada Todd A Drumm Department of Mathematics Howard University Washington DC 20059 USA William M Goldman Department of Mathematics University of Maryland College Park MD 20742 USA