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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The geometry of flip graphs and mapping class groups
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by Valentina Disarlo and Hugo Parlier PDF
Trans. Amer. Math. Soc. 372 (2019), 3809-3844


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:
  • Hugo Parlier
  • Affiliation: Mathematics Research Unit, University of Luxembourg, 4365 Esch-zur-Alzette, Luxembourg
  • Email:
  • 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).
  • © Copyright 2019 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:
  • MathSciNet review: 4009420