A geometric boundary for the moduli space of grafted surfaces

Abstract

Let $S$ be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space $\mathcal{PT}(S)$ of complex projective structures, with a boundary consisting of projective classes of half-translation surfaces. Thurston established an equivalence between complex projective structures and hyperbolic surfaces grafted along a measured lamination, leading to a homeomorphism $\mathcal{PT}(S) \cong \mathcal{T}(S) \times \mathcal{ML}(S)$. Our bordification is geometric in the sense that convergence to points on the boundary corresponds to the geometric convergence of grafted surfaces to half-translation surfaces (up to rescaling). This result relies on recent work by Calderon and Farre on the orthogeodesic foliation construction. Finally, we introduce a change of perspective, viewing grafted surfaces as a deformation (which we term "inflation") of half-translation surfaces, consisting of inserting negatively curved regions.