Cubical geometry in the polygonalisation complex
We 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...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
July 2019
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| In: |
Mathematical proceedings of the Cambridge Philosophical Society
Year: 2019, Volume: 167, Issue: 1, Pages: 1-22 |
| ISSN: | 1469-8064 |
| DOI: | 10.1017/S0305004118000130 |
| Online Access: | Verlag, Volltext: https://doi.org/10.1017/S0305004118000130 Verlag: https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/cubical-geometry-in-the-polygonalisation-complex/329A12E1BAB737B5FD361ACC4A573D23 
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| Author Notes: | Mark C. Bell, Valentina Disarlo, Robert Tang |
| Summary: | We 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. |
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| Item Description: | Gesehen am 15.10.2019 |
| Physical Description: | Online Resource |
| ISSN: | 1469-8064 |
| DOI: | 10.1017/S0305004118000130 |