Regenerating singular hyperbolic structures from Sol

Michael Heusener (1) , Joan Porti (2) , Eva Suarez (3)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 UAB - Departament de Matemàtiques [Barcelona]
3 Mathematisches Institut [Tübingen]

Let M be a torus bundle over the circle with an orientation preserving Anosov monodromy. The manifold M admits a geometric structure modeled on Sol. We prove that the Sol structure can be deformed into singular hyperbolic cone structures whose singular locus Σ ⊂ M is the mapping torus of the fixed point of the monodromy. The hyperbolic cone metrics are parametred by the cone angle α in the interval (0, 2π). When α → 2π, the cone manifolds collapse to the basis circle of the fibration, and they can be rescaled in the direction of the fibers to converge to the Sol manifold.

Michael Heusener, Joan Porti, Eva Suarez. Regenerating singular hyperbolic structures from Sol. Journal of Differential Geometry, 2001, pp.439-478. ⟨hal-00465252⟩