Strip deformations of decorated hyperbolic polygons

Abstract

In this paper, we study the hyperbolic and parabolic strip deformations of ideal (possibly once-punctured) hyperbolic polygons whose vertices are decorated with horoballs. We prove that the interiors of their arc complexes parametrise the open convex set of all uniformly lengthening infinitesimal deformations of the decorated hyperbolic metrics on these surfaces, motivated by the work of Danciger–Guéritaud–Kassel. We also give a version of this result for the undecorated ideal polygons and once-punctured ideal polygons.