Hyperbolic structures from Sol on pseudo-Anosov mapping tori

TL;DR

This paper explores how pseudo-Anosov monodromies induce Sol structures on mapping tori and demonstrates their deformation into cone hyperbolic structures with adjustable angles.

Contribution

It establishes conditions under which Sol structures from pseudo-Anosov maps can be deformed into cone hyperbolic structures with decreasing cone angles.

Findings

Sol structures arise from invariant foliations of pseudo-Anosov maps.

Deformation into cone hyperbolic structures is possible under specific eigenvalue conditions.

Cone angles can be decreased from multiples of 2π during deformation.

Abstract

The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $\phi:S\rightarrow S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on the closed surface, then the Sol structure can be deformed to nearby cone hyperbolic structures, in the sense of projective structures. The cone angles can be chosen to be decreasing from multiples of $2\pi$.