Affine deformations of a three-holed sphere

Details Affine deformations of a three-holed sphere Affine deformations of a three-holed sphere

2009-07-03

arxiv.org/abs/0907.0690v1

Geometry & Topology 14 (2010), 1355-1382

Mathematics

Differential Geometry

30 pages, 7 figures

Scientific paper

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such 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(4,Z).

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