Parametrisation of the deformation spaces of ideal polygons and punctured polygon
To every hyperbolic surface, one can associate a simplicial complex called the arc complex whose 0-th skeleton is the set of isotopy classes of non-trivial embedded arcs and there is a k-simplex for every (k+1)-tuple of pairwise disjoint and distinct isotopy classes. The arc complexes of ideal polygons and punctured polygons are finite and are homeomorphic to spheres. The deformation spaces of these two hyperbolic surfaces can be completely parametrised via their arc complexes using strip deformations along the arcs. This is a partial generalisation of a result by Dancinger-Guéritaud-Kassel.