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Regenerating singular hyperbolic structures from Sol. (English) Zbl 1042.57008
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 $\Bbb R^2$ and hence fixes the projection of the origin in $\Bbb R^2$). 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 $sl_2(\Bbb C)$ 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.

57M50Geometric structures on low-dimensional manifolds
57N10Topology of general $3$-manifolds