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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

The geometry of flip graphs and mapping class groups


Authors: Valentina Disarlo and Hugo Parlier
Journal: Trans. Amer. Math. Soc. 372 (2019), 3809-3844
MSC (2010): Primary 05C25, 30F60, 32G15, 57M50; Secondary 05C12, 05C60, 30F10, 57M07, 57M60.
DOI: https://doi.org/10.1090/tran/7356
Published electronically: June 17, 2019
MathSciNet review: 4009420
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Abstract: The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichmüller space and is quasi-isometric to the underlying mapping class group. We study this space in two main directions. We first show that strata corresponding to triangulations containing a same multiarc are strongly convex within the whole space and use this result to deduce properties about the mapping class group. We then focus on the quotient of this space by the mapping class group to obtain a type of combinatorial moduli space. In particular, we are able to identity how the diameters of the resulting spaces grow in terms of the complexity of the underlying surfaces.


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Additional Information

Valentina Disarlo
Affiliation: Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Germany
MR Author ID: 983316
Email: vdisarlo@mathi.uni-heidelberg.de

Hugo Parlier
Affiliation: Mathematics Research Unit, University of Luxembourg, 4365 Esch-zur-Alzette, Luxembourg
Email: hugo.parlier@uni.lu

Keywords: Flip graphs, triangulations of surfaces, combinatorial moduli spaces, mapping class groups
Received by editor(s): October 31, 2016
Received by editor(s) in revised form: April 18, 2017
Published electronically: June 17, 2019
Additional Notes: Research of the first author was partially funded by an International Scholarship from the University of Fribourg. Part of this work was carried out while the first author was visiting the second author at the University of Fribourg. She is grateful to the department and the staff for the warm hospitality. She also acknowledges the support of Indiana University Provost’s Travel Award for Women in Science.
Research of the second author was supported by Swiss National Science Foundation grants numbers PP00P2_15302 and PP00P2_128557.\endgraf The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
Article copyright: © Copyright 2019 Valentina Disarlo and Hugo Parlier