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Regenerating singular hyperbolic structures from Sol. (English) Zbl 1042.57008
The mapping torus of an orientation preserving Anosov homeomorphism φ of the 2-torus (i.e., a compact 3-manifold fibering over the circle with fiber a torus T 2 ) admits a geometric structure modeled on the solvable geometry Sol. There is a natural section ΣM of the fibration MS 1 (φ:T 2 T 2 lifts to a linear map of 2 and hence fixes the projection of the origin in 2 ). The main result of the paper states that there exists a family of hyperbolic cone structures on M with singular set Σ parametrized by the cone angle α(0,2π). When α2π this family collapses to a circle (the basis of the fibration MS 1 ); also, the metrics can be rescaled in the direction of the fibers so that they converge to the Sol structure on M (when α0 this family of cone manifolds converges to the complete hyperbolic structure on MΣ). An explicit construction of the deformations of the Sol structure on M is given by using the Cartan splitting of the Lie algebra sl 2 () and associated Killing fields, and also a result about algebraic deformations of reducible representations proved in a previous paper by the authors [J. Reine Angew. Math. 530, 191–227 (2001; Zbl 0964.57006)]. An example of such a manifold M is obtained by 0-surgery on the figure eight knot which has been considered by Jorgensen and Thurston; also, Hilden, Lozano and Montesinos constructed an explicit family of Dirichlet polyhedra for this manifold collapsing to a segment (whose ends are identified to give S 1 ), and the third named author of the present paper showed that this family of polyhedra can be rescaled to converge to a Sol structure.
MSC:
57M50Geometric structures on low-dimensional manifolds
57N10Topology of general 3-manifolds