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arXiv · 2505.01285

Polyhedral realisations of finite arc complexes using strip deformations

François Guéritaud·Pallavi Panda
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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.

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François Guéritaud, Pallavi Panda. 2025-05-02. Polyhedral realisations of finite arc complexes using strip deformations. https://arxiv.org/abs/2505.01285

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