Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-04-25T06:58:39.797Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigidity of mapping class groups MOD powers of twists

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  07 April 2025

Giorgio Mangioni Open the ORCID record for Giorgio Mangioni [Opens in a new window]  and
Alessandro Sisto Open the ORCID record for Alessandro Sisto [Opens in a new window]
Giorgio Mangioni
Affiliation:
Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom (gm2070@hw.ac.uk)
Alessandro Sisto*
Affiliation:
Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom (a.sisto@hw.ac.uk) (corresponding author)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov’s theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result to prove quasi-isometric rigidity of these quotients, answering a question of Behrstock, Hagen, Martin, and Sisto in the case of punctured spheres. Finally, we show that the automorphism groups of our quotients of mapping class groups are “small”, as are their abstract commensurators. This is again an analogue of a theorem of Ivanov about the automorphism group of the mapping class group.

In the process, we develop techniques to extract combinatorial data from a quasi-isometry of a hierarchically hyperbolic space, and use them to give a different proof of a result of Bowditch about quasi-isometric rigidity of pants graphs of punctured spheres.

Keywords

mapping class groupquasi-isometric rigidityIvanov’s TheoremDehn twist quotients

MSC classification

Primary: 20F65: Geometric group theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 71
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Agol, I.. The virtual Haken conjecture. In Groves, D., and Manning, J., Eds. Doc. Math. Vol. 18, (EMS press, Bielefeld, 2013). With an appendix by Agol.Google Scholar
Antolín, Y., Minasyan, A. and Sisto, A.. Commensurating endomorphisms of acylindrically hyperbolic groups and applications. Groups Geom. Dyn. 10 (2016), 1149–1210.CrossRefGoogle Scholar
Aramayona, J. and Leiniger, C. J.. Finite rigid sets in curve complexes. J. Topol. Anal. 5 (2013), 183–203.CrossRefGoogle Scholar
Aramayona, J., and Souto, J.. Rigidity phenomena in the mapping class group. In Papadopoulos, A. (Ed.), Handbook of Teichmüller theory, Vol. VI, (EMS press, Bielefeld, 2016).Google Scholar
Behrstock, J., Hagen, M. F. and Sisto, A.. Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups. Proc. Lond. Math. Soc. (3). 114 (2017), 890–926.CrossRefGoogle Scholar
Behrstock, J., Hagen, M. F. and Sisto, A.. Hierarchically hyperbolic spaces II: Combination theorems and the distance formula. Pacific J. Math. 299 (2019), 257–338.CrossRefGoogle Scholar
Behrstock, J., Hagen, M. F. and Sisto, A.. Quasiflats in hierarchically hyperbolic spaces. Duke Math. J. 170 (2021), 909–996.CrossRefGoogle Scholar
Behrstock, J., Hagen, M., Martin, A. and Sisto, A.. A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups. J. Topol. 17 (2024), 94 Id/No e12351.CrossRefGoogle Scholar
Behrstock, J., Kleiner, B., Minsky, Y. and Mosher, L.. Geometry and rigidity of mapping class groups. Geom. Topol. 16 (2012), 781–888.CrossRefGoogle Scholar
Bestvina, M., Bromberg, K. and Fujiwara, K.. Constructing group actions on quasi-trees and applications to mapping class groups. Publ. Math. Inst. Hautes Études Sci. 122 (2015), 1–64.CrossRefGoogle Scholar
Bestvina, M. and Fujiwara, K.. Bounded cohomology of subgroups of mapping class groups. Geom. Topol. 6 (2002), 69–89.CrossRefGoogle Scholar
Bowditch, B. H.. Rigidity of the strongly separating curve graph. Michigan Math. J. 65 (2016), 813–832.CrossRefGoogle Scholar
Bowditch, B. H.. Large-scale rank and rigidity of the Weil-Petersson metric. Groups Geom. Dyn. 14 (2020), 607–652.CrossRefGoogle Scholar
Christopher Hruska, G.. Relative hyperbolicity and relative quasiconvexity for countable group. Algebr. Geom. Topol. 10 (2010), 1807–1856.CrossRefGoogle Scholar
Clay, M. and Mangahas, J.. Hyperbolic quotients of projection complexes. Groups Geom. Dyn. 16 (2022), 225–246.CrossRefGoogle Scholar
Coulon, R. and Sela, Z.. Equations in burnside groups (2021), arXiv preprint arXiv:2112.07409.Google Scholar
Dahmani, F.. The normal closure of big Dehn twists and plate spinning with rotating families. Geom. Topol. 22 (2018), 4113–4144.CrossRefGoogle Scholar
Dahmani, F., Guirardel, V. and Osin, D.. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Mem. Am. Math. Soc. 245 (2017), .Google Scholar
Dahmani, F., Hagen, M. F. and Sisto, A.. Dehn filling Dehn twists. Proc. Roy. Soc. Edinburgh Sect. A. 151 (2021), 28–51.CrossRefGoogle Scholar
Dehn, M.. Papers on Group Theory and topology. Translated from the German and with introductions and an appendix by John Stillwell, With an appendix by Otto Schreier (Springer-Verlag, New York, 1987).Google Scholar
Farb, B. and Margalit, D.. A primer on mapping class groups. Of Princeton Mathematical Series., Vol. 49 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Farb, B. and Mosher, L.. Convex cocompact subgroups of mapping class groups. Geom. Topol. 6 (2002), 91–152.CrossRefGoogle Scholar
Funar, L.. On the TQFT representations of the mapping class groups. Pacific J. Math. 188 (1999), 251–274.CrossRefGoogle Scholar
Hamenstaedt, U.. Geometry of the mapping class groups III: quasi-isometric rigidity (2005).Google Scholar
Korkmaz, M.. Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topol. Appl. 95 (1999), 85–111.Google Scholar
Masur, H. A. and Minsky, Y. N.. Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal. 10 (2000), 902–974.Google Scholar
Nielsen, J.. Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta Math. 50 (1927), 189–358.CrossRefGoogle Scholar
Ivanov, N. V.. Automorphisms of complexes of curves and of Teichmüller spaces. Travaux en Cours, Vol.56, (Hermann, Paris, 1997).Google Scholar
Wade, R. C. H.. Uniform bounds for bounded geodesic image theorems. J. Reine Angew. Math. 709 (2015), 219–228.Google Scholar
Kent IV, R. P. and C. J. Leiniger, Should be. Shadows of mapping class groups: capturing convex cocompactness. Geom. Funct. Anal. 18 (2008), 1270–1325.CrossRefGoogle Scholar
Schwartz, R. E.. The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. 82 (1996), 133–168.CrossRefGoogle Scholar