The arc complexes of partially decorated hyperbolic polygons
Combinatorics
2025-02-25 v2 Geometric Topology
Abstract
We consider two families of hyperbolic polygons: ideal and ideal once-punctured, some of whose spikes are decorated with horoballs. We show that the arc complexes of these two families of surfaces, generated by edge-to-edge arcs and edge-to-decorated-spike arcs, are closed piecewise linear balls. This is proved in a completely combinatorial setting: compact polygons whose vertices are assigned red or blue colouring. In order to prove the ballness, we show that these simplicial complexes are pseudo-manifolds and use shellability to conclude. As a consequence, we parametrise weakly-lengthening deformations of the partially decorated hyperbolic polygons.
Cite
@article{arxiv.2306.06695,
title = {The arc complexes of partially decorated hyperbolic polygons},
author = {Pallavi Panda},
journal= {arXiv preprint arXiv:2306.06695},
year = {2025}
}