Deformations of the modular group as a quasifuchsian correspondence

Abstract:

When viewed as a $(2:2)$ holomorphic correspondence on the Riemann sphere, the modular group $PSL_2({\mathbb Z})$ has a moduli space ${\mathcal Q}$ of non-trivial deformations for which the limit set remains a topological circle. This space is analogous to a Bers slice of the deformation space of a Fuchsian group as a Kleinian group, but there are certain differences. A Bers slice contains a single quasiconformal conjugacy class of Kleinian groups: we show that for an open dense set of parameter values in ${\mathcal Q}$ the correspondence belongs to a single quasi-conformal conjugacy class, but that at a countable set ${\mathcal C}$ of isolated parameter values it satisfies an additional critical relation. We classify these relations, propose ‘pleating coordinates’ for ${\mathcal Q}$, and investigate how the correspondence degenerates on the boundary of ${\mathcal Q}$. In particular, we show that there is a point on the boundary of ${\mathcal Q}$ where the correspondence degenerates into a mating between $PSL_2(\mathbb Z)$ and the quadratic polynomial $z \to z^2+1/4$. A key ingredient in our analysis is a bijection between ${\mathcal Q}\setminus {\mathcal C}$ and an intermediate cover between the moduli space of the space of non-critical grand orbits of the correspondence, and its universal cover, the corresponding Teichmüller space.