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Cubical geometry in the polygonalisation complex

Published online by Cambridge University Press:  08 May 2018

MARK C. BELL ,
VALENTINA DISARLO  and
ROBERT TANG
MARK C. BELL
Affiliation:
Warwick Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, U.K. e-mail: m.bell.7@warwick.ac.uk
VALENTINA DISARLO
Affiliation:
Mathematisches Institut, Heidelberg Universitaet, Heidelberg 691220, Germany. e-mail: vdisarlo@mathi.uni-heidelberg.de
ROBERT TANG
Affiliation:
Topology and Geometry of Manifolds Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan. e-mail: robert.tang@oist.jp
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 1 - 22
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

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