Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T07:11:36.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Surface groups, infinite generating sets, and stable commutator length

Published online by Cambridge University Press:  23 April 2019

Dan Margalit  and
Andrew Putman
Dan Margalit
Affiliation:
Georgia Institute of Technology, School of Mathematics, 686 Cherry St, Atlanta, GA30306, USA ([email protected])
Andrew Putman
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN46556, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

Keywords

Surface groupstable commutator lengthmapping class group
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2379 - 2386
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

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.)

References

1Bestvina, M., Bromberg, K. and Fujiwara, K.. Stable commutator length on mapping class groups. Ann. Inst. Fourier (Grenoble) 66 (2016), 871–898.CrossRefGoogle Scholar
2Brandenbursky, M. and Marcinkowski, M.. Aut-invariant norms and Aut-invariant quasimorphisms on free and surface group, to appear in Comment. Math. Helv.Google Scholar
3Calegari, D.. Word length in surface groups with characteristic generating sets. Proc. Amer. Math. Soc. 136 (2008), 2631–2637.CrossRefGoogle Scholar
4Calegari, D.. scl. MSJ Memoirs,vol. 20 (Tokyo: Mathematical Society of Japan, 2009).CrossRefGoogle Scholar
5Calegari, D. and Fujiwara, K.. Stable commutator length in word-hyperbolic groups. Groups Geom. Dyn. 4 (2010), 59–90.CrossRefGoogle Scholar
6Clay, A. and Rolfsen, D.. Ordered groups and topology. Graduate Studies in Mathematics,vol. 176 (Providence, RI: American Mathematical Society, 2016).CrossRefGoogle Scholar
7Farb, B. and Margalit, D.. A primer on mapping class groups. Princeton Mathematical Series,vol. 49 (Princeton, NJ: Princeton University Press, 2012).Google Scholar
8Hass, J. and Scott, P.. Intersections of curves on surfaces. Israel J. Math. 51 (1985), 90–120.CrossRefGoogle Scholar
9Hass, J. and Scott, P.. Shortening curves on surfaces. Topology 33 (1994), 25–43.CrossRefGoogle Scholar
10Kra, I.. On the Nielsen–Thurston–Bers type of some self-maps of Riemann surfaces. Acta Math. 146 (1981), 231–270.CrossRefGoogle Scholar
11Paterson, J. M.. A combinatorial algorithm for immersed loops in surfaces. Topol. Appl. 123 (2002), 205–234.CrossRefGoogle Scholar
12Putman, A.. An infinite presentation of the Torelli group. Geom. Funct. Anal. 19 (2009), 591–643.CrossRefGoogle Scholar