A geometric compactification of the moduli space of grafted surfaces

Abstract

In this thesis, we study the degenerations of complex projective structures on an orientable surface S of genus at least two, aiming to describe a compactification of their moduli space and provide a geometric interpretation of the boundary points. The moduli space &#119980;&#119983;(S) of complex projective structures admits a parametrisation due to Thurston via grafting: each structure corresponds to a metric on S that is obtained from a hyperbolic one by grafting, namely inserting, flat parts along a measured lamination. This construction yields a homeomorphism &#119980;&#119983;(S) &cong; &#119983;(S) &times; &#8499;&#8466;(S), where &#119983;(S) is Teichm&uuml;ller space and &#8499;&#8466;(S) the space of measured laminations. We refer to the metric surfaces resulting from grafting as grafted surfaces. <br /> We prove that degenerating sequences of grafted surfaces, suitably rescaled, can converge geometrically to half-translation surfaces, that is, Euclidean surfaces with cone singularities. We use the orthogeodesic foliation introduced by Calderon and Farre to analyse this phenomenon, and we construct a bordification of &#119980;&#119983;(S) whose boundary at infinity is given by the moduli space &#8473;_&#8450;&#119980;&#119983;(S) of half-translation surfaces up to rotation and rescale. The topology on this bordification is the one induced by a marked version of Gromov-Hausdorff convergence introduced in this work. <br /> We also show that &#119980;&#119983;(S) embeds into the space of projective geodesic currents and this embedding extends continuously to our bordification too. We describe the whole closure of its image, when embedding &#119980;&#119983;(S) into projective currents using a novel l¹-variant of the grafted metric. The boundary in this case is described by mixed structures, which have appeared in different forms in the bordification of other moduli spaces of geometric structures on surfaces. As an application, we describe the limits of so-called generalised stretch rays in Teichm&uuml;ller space.