Regenerating Singular Hyperbolic Structures from Sol

Abstract

Let M be a torus bundle over S¹ 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 of the fibration S¹, and they can be rescaled in the direction of the fibers to converge to the Sol manifold.