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Regenerating singular hyperbolic structures from Sol. (English) Zbl 1042.57008
The mapping torus of an orientation preserving Anosov homeomorphism $\varphi$ 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 $M\to {S}^{1}$ ($\varphi :{T}^{2}\to {T}^{2}$ lifts to a linear map of ${ℝ}^{2}$ and hence fixes the projection of the origin in ${ℝ}^{2}$). 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 $\alpha \in \left(0,2\pi \right)$. 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 $s{l}_{2}\left(ℂ\right)$ 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:
 57M50 Geometric structures on low-dimensional manifolds 57N10 Topology of general 3-manifolds