## 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

## 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.## References

- Ian Agol,
*Ideal triangulations of pseudo-Anosov mapping tori*, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 1–17. MR**2866919**, DOI 10.1090/conm/560/11087 - Javier Aramayona, Thomas Koberda, and Hugo Parlier,
*Injective maps between flip graphs*, Ann. Inst. Fourier (Grenoble)**65**(2015), no. 5, 2037–2055 (English, with English and French summaries). MR**3449205**, DOI 10.5802/aif.2981 - J. Aramayona, C. Lecuire, H. Parlier, and K. J. Shackleton,
*Convexity of strata in diagonal pants graphs of surfaces*, Publ. Mat.**57**(2013), no. 1, 219–237. MR**3058933**, DOI 10.5565/PUBLMAT_{5}7113_{0}8 - Javier Aramayona, Hugo Parlier, and Kenneth J. Shackleton,
*Totally geodesic subgraphs of the pants complex*, Math. Res. Lett.**15**(2008), no. 2, 309–320. MR**2385643**, DOI 10.4310/MRL.2008.v15.n2.a9 - Javier Aramayona, Hugo Parlier, and Kenneth J. Shackleton,
*Constructing convex planes in the pants complex*, Proc. Amer. Math. Soc.**137**(2009), no. 10, 3523–3531. MR**2515421**, DOI 10.1090/S0002-9939-09-09907-9 - Prosenjit Bose and Sander Verdonschot,
*A history of flips in combinatorial triangulations*, Computational geometry, Lecture Notes in Comput. Sci., vol. 7579, Springer, Cham, 2011, pp. 29–44. MR**3446116**, DOI 10.1007/978-3-642-34191-5_{3} - William G. Brown,
*Enumeration of triangulations of the disk*, Proc. London Math. Soc. (3)**14**(1964), 746–768. MR**168485**, DOI 10.1112/plms/s3-14.4.746 - W. Cavendish, Growth of the diameter of the pants graph modulo the mapping class group, preprint (2011).
- William Cavendish and Hugo Parlier,
*Growth of the Weil-Petersson diameter of moduli space*, Duke Math. J.**161**(2012), no. 1, 139–171. MR**2872556**, DOI 10.1215/00127094-1507312 - C. Cortés, C. I. Grima, F. Hurtado, A. Márquez, F. Santos, and J. Valenzuela,
*Transforming triangulations on nonplanar surfaces*, SIAM J. Discrete Math.**24**(2010), no. 3, 821–840. MR**2680217**, DOI 10.1137/070697987 - Benson Farb and Dan Margalit,
*A primer on mapping class groups*, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR**2850125** - Sergey Fomin, Michael Shapiro, and Dylan Thurston,
*Cluster algebras and triangulated surfaces. I. Cluster complexes*, Acta Math.**201**(2008), no. 1, 83–146. MR**2448067**, DOI 10.1007/s11511-008-0030-7 - S. Fomin and D. Thurston, Cluster algebras and triangulated surfaces. Part II: Lambda lengths, preprint (2012).
- U. Hamenstädt, Geometry of the mapping class group II: A biautomatic structure, preprint (2009).
- Allen Hatcher,
*On triangulations of surfaces*, Topology Appl.**40**(1991), no. 2, 189–194. MR**1123262**, DOI 10.1016/0166-8641(91)90050-V - Mustafa Korkmaz and Athanase Papadopoulos,
*On the ideal triangulation graph of a punctured surface*, Ann. Inst. Fourier (Grenoble)**62**(2012), no. 4, 1367–1382 (English, with English and French summaries). MR**3025746**, DOI 10.5802/aif.2725 - Jesús A. De Loera, Jörg Rambau, and Francisco Santos,
*Triangulations*, Algorithms and Computation in Mathematics, vol. 25, Springer-Verlag, Berlin, 2010. Structures for algorithms and applications. MR**2743368**, DOI 10.1007/978-3-642-12971-1 - H. A. Masur and Y. N. Minsky,
*Geometry of the complex of curves. II. Hierarchical structure*, Geom. Funct. Anal.**10**(2000), no. 4, 902–974. MR**1791145**, DOI 10.1007/PL00001643 - Lee Mosher,
*Tiling the projective foliation space of a punctured surface*, Trans. Amer. Math. Soc.**306**(1988), no. 1, 1–70. MR**927683**, DOI 10.1090/S0002-9947-1988-0927683-0 - Lee Mosher,
*Mapping class groups are automatic*, Ann. of Math. (2)**142**(1995), no. 2, 303–384. MR**1343324**, DOI 10.2307/2118637 - Seiya Negami,
*Diagonal flips in triangulations on closed surfaces, estimating upper bounds*, Yokohama Math. J.**45**(1998), no. 2, 113–124. MR**1637454** - Seiya Negami,
*Diagonal flips in pseudo-triangulations on closed surfaces*, Discrete Math.**240**(2001), no. 1-3, 187–196. MR**1855053**, DOI 10.1016/S0012-365X(00)00391-5 - R. C. Penner,
*Weil-Petersson volumes*, J. Differential Geom.**35**(1992), no. 3, 559–608. MR**1163449**, DOI 10.4310/jdg/1214448257 - R. C. Penner,
*Universal constructions in Teichmüller theory*, Adv. Math.**98**(1993), no. 2, 143–215. MR**1213724**, DOI 10.1006/aima.1993.1015 - Robert C. Penner,
*Decorated Teichmüller theory*, QGM Master Class Series, European Mathematical Society (EMS), Zürich, 2012. With a foreword by Yuri I. Manin. MR**3052157**, DOI 10.4171/075 - Lionel Pournin,
*The diameter of associahedra*, Adv. Math.**259**(2014), 13–42. MR**3197650**, DOI 10.1016/j.aim.2014.02.035 - Kasra Rafi and Saul Schleimer,
*Covers and the curve complex*, Geom. Topol.**13**(2009), no. 4, 2141–2162. MR**2507116**, DOI 10.2140/gt.2009.13.2141 - Kasra Rafi and Jing Tao,
*The diameter of the thick part of moduli space and simultaneous Whitehead moves*, Duke Math. J.**162**(2013), no. 10, 1833–1876. MR**3079261**, DOI 10.1215/00127094-2323128 - Kasra Rafi and Jing Tao,
*Uniform growth rate*, Proc. Amer. Math. Soc.**144**(2016), no. 4, 1415–1427. MR**3451220**, DOI 10.1090/proc/12816 - Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston,
*Rotation distance, triangulations, and hyperbolic geometry*, J. Amer. Math. Soc.**1**(1988), no. 3, 647–681. MR**928904**, DOI 10.1090/S0894-0347-1988-0928904-4 - Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston,
*Short encodings of evolving structures*, SIAM J. Discrete Math.**5**(1992), no. 3, 428–450. MR**1172751**, DOI 10.1137/0405034 - James Dillon Stasheff,
*Homotopy associativity of $H$-spaces. I, II*, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid.**108**(1963), 293–312. MR**0158400**, DOI 10.1090/S0002-9947-1963-0158400-5 - Dov Tamari,
*Monoïdes préordonnés et chaînes de Malcev*, Université de Paris, Paris, 1951 (French). Thèse. MR**0051833** - Samuel J. Taylor and Alexander Zupan,
*Products of Farey graphs are totally geodesic in the pants graph*, J. Topol. Anal.**8**(2016), no. 2, 287–311. MR**3474077**, DOI 10.1142/S1793525316500096 - Yasuyuki Tsukui,
*Transformations of cubic graphs*, J. Franklin Inst. B**333**(1996), no. 4, 565–575. MR**1401212**, DOI 10.1016/0016-0032(96)00015-4

## 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
- 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: https://doi.org/10.1090/tran/7356
- MathSciNet review: 4009420