Regenerating singular hyperbolic structures from Sol
The mapping torus of an orientation preserving Anosov homeomorphism \(\phi\) 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 \(\Sigma \subset M\) of the fibration \(M \to S^1\) (\(\phi:T^2 \to T^2\) lifts to a linear map of \(\mathbb R^2\) and hence fixes the projection of the origin in \(\mathbb R^2\)). The main result of the paper states that there exists a family of hyperbolic cone structures on \(M\) with singular set \(\Sigma\) parametrized by the cone angle \(\alpha \in (0,2\pi)\). When \(\alpha \to 2\pi\) this family collapses to a circle (the basis of the fibration \(M \to S^1\)); also, the metrics can be rescaled in the direction of the fibers so that they converge to the Sol structure on \(M\) (when \(\alpha \to 0\) this family of cone manifolds converges to the complete hyperbolic structure on \(M \setminus \Sigma\)). An explicit construction of the deformations of the Sol structure on \(M\) is given by using the Cartan splitting of the Lie algebra \(sl_2(\mathbb C)\) 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.
scientific article; zbMATH DE number 2053205