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arXiv · 2306.06695

The arc complexes of partially decorated hyperbolic polygons

Pallavi Panda
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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.

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Pallavi Panda. 2023-06-11. The arc complexes of partially decorated hyperbolic polygons. https://arxiv.org/abs/2306.06695

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