Hyperbolic structures from Sol on pseudo-Anosov mapping tori (Q258893)

Let \(\phi: S\to S\) be a pseudo-Anosov diffeomorphism of a surface \(S\) (possibly with punctures), with invariant singular foliations \({\mathcal F}^s,{\mathcal F}^u\), transverse measures \(\mu^s,\mu^u\), and dilatation factor \(\lambda\). Let \(M_\phi\) be the mapping torus of \(\phi\).NEWLINENEWLINERecall that the Lie group \(Sol\) is \({\mathbb R}^3\) with the metric \(e^{2t}dx^2+e^{-2t}dy^2+dt^2\). There is a natural singular Sol structure on \(M_\phi\) whose singular locus \(\Sigma\) is given by the mapping torus of the union of the singularities of \({\mathcal F}^s,{\mathcal F}^u\) and the punctures. Let \(N_\phi=M_\phi\setminus \Sigma\) and \(\Gamma=\pi_1N_\phi\).NEWLINENEWLINESol contains ``vertical'' hyperbolic planes and in the singular Sol structure on \(M_\phi\), these can be seen as products of a leaf of the singular foliation \({\mathcal F}^s\) with the \(S^1\) direction. Projecting along \({\mathcal F}^u\) onto one of the hyperbolic planes provides \(M_\phi\) with the structure of a ``transversely hyperbolic foliation'' and an associated metabelian representation \(\rho_0:\Gamma\to \mathrm{PSL}(2,{\mathbb R})\) where the generators of \(\pi_1S\) are sent to an upper triangular matrix with upper right entry equal to the value of \(\mu^u\) on that generator (in particular, loops around punctures are sent to the identity) and the extending element \(\tau\) corresponding to the \(S^1\) direction is sent to a diagonal matrix with entries \(\sqrt{\lambda}\) and \(\frac{1}{\sqrt{\lambda}}\).NEWLINENEWLINEUnder the standing assumption that the invariant foliations are orientable and that \(\phi^*: H^1(S)\to H^1(S)\) does not have \(1\) as an eigenvalue, the paper under review proves that \(\rho_0\) is a smooth point of the representation variety and that its local dimension is \(3+\sharp\left\{\text{components of }\partial N_\phi\right\}\).NEWLINENEWLINEThis smoothness result is then used to find representations near \(\rho_0\). The author proves that there exists a family of singular hyperbolic structures on \(M_\phi\), smooth on \(N_\phi\), that degenerates to the transversely hyperbolic foliation. Further he is able to control the singularities in order to obtain a family of hyperbolic cone manifolds.NEWLINENEWLINEIn a final section a genus \(2\) example is computed in full detail.