Author:
BELL MARK C.,DISARLO VALENTINA,TANG ROBERT
Abstract
AbstractWe introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface.We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, different surfaces have different polygonalisation complexes. The polygonalisation complex is not CAT(0), but we can characterise the vertices where Gromov's link condition fails. This gives a tool for proving that, generically, the automorphism group of the polygonalisation complex is the (extended) mapping class group of the surface.
Publisher
Cambridge University Press (CUP)
Cited by
4 articles.
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1. Rigidity of the saddle connection complex;Journal of Topology;2022-06-30
2. Finite rigid sets in flip graphs;Transactions of the American Mathematical Society;2021-12-02
3. Finite rigid sets in arc complexes;Algebraic & Geometric Topology;2020-12-08
4. Eccentricities in the flip‐graphs of convex polygons;Journal of Graph Theory;2019-01-17