Polyhedral realisations of finite arc complexes using strip deformations

We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of deformations. We describe this cone in terms of the arc complex of the surface: specifically, this paper focuses on the surfaces for which that complex is finite. Those surfaces form four families: (ideal) polygons, once-punctured polygons, one-holed polygons (or crowns’'), and Möbius strips with spikes. In each case, we describe a natural simplicial decomposition of the projectivised admissible cone and of each of its faces, realizing them as appropriate arc complexes.