Polyhedral realisations of finite arc complexes using strip deformations
Differential Geometry
2025-05-05 v1 Combinatorics
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.
Cite
@article{arxiv.2505.01285,
title = {Polyhedral realisations of finite arc complexes using strip deformations},
author = {François Guéritaud and Pallavi Panda},
journal= {arXiv preprint arXiv:2505.01285},
year = {2025}
}