Polyhedral realisations of finite arc complexes using strip deformations

TL;DR

This paper explores how to geometrically realize finite arc complexes of certain hyperbolic surfaces through strip deformations, providing explicit descriptions of the admissible deformation cones in terms of arc complexes.

Contribution

It introduces a novel geometric realization of admissible deformation cones for specific hyperbolic surfaces using arc complexes, focusing on finite cases.

Findings

Describes the admissible deformation cone as a convex cone related to arc complexes.

Provides a simplicial decomposition of the projectivized cone for various surface types.

Realizes the cones and their faces explicitly as arc complexes.

Abstract

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 M\"obius 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.