Stable commutator length in Baumslag–Solitar groups and quasimorphisms for tree actions
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- by Matt Clay, Max Forester and Joel Louwsma PDF
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Abstract:
This paper has two parts: on Baumslag–Solitar groups and on general $G$–trees.
In the first part we establish bounds for stable commutator length (scl) in Baumslag–Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces.
In the second part we establish a universal lower bound of $1/12$ for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group $BS(2,3)$ show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions.
Returning to Baumslag–Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval $(0, 1/12)$.
References
- Christophe Bavard, Longueur stable des commutateurs, Enseign. Math. (2) 37 (1991), no. 1-2, 109–150 (French). MR 1115747
- Mladen Bestvina, Ken Bromberg, and Koji Fujiwara, Stable commutator length on mapping class groups, preprint, arXiv:1306.2394v1.
- Mladen Bestvina and Koji Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89. MR 1914565, DOI 10.2140/gt.2002.6.69
- Noel Brady, Matt Clay, and Max Forester, Turn graphs and extremal surfaces in free groups, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 171–178. MR 2866930, DOI 10.1090/conm/560/11098
- Robert Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 53–63. MR 624804
- Danny Calegari, scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, 2009. MR 2527432, DOI 10.1142/e018
- Danny Calegari, Stable commutator length is rational in free groups, J. Amer. Math. Soc. 22 (2009), no. 4, 941–961. MR 2525776, DOI 10.1090/S0894-0347-09-00634-1
- Danny Calegari, scl, sails, and surgery, J. Topol. 4 (2011), no. 2, 305–326. MR 2805993, DOI 10.1112/jtopol/jtr001
- Danny Calegari and Koji Fujiwara, Stable commutator length in word-hyperbolic groups, Groups Geom. Dyn. 4 (2010), no. 1, 59–90. MR 2566301, DOI 10.4171/GGD/75
- Donald J. Collins, On embedding groups and the conjugacy problem, J. London Math. Soc. (2) 1 (1969), 674–682. MR 252489, DOI 10.1112/jlms/s2-1.1.674
- Andrew J. Duncan and James Howie, The genus problem for one-relator products of locally indicable groups, Math. Z. 208 (1991), no. 2, 225–237. MR 1128707, DOI 10.1007/BF02571522
- David B. A. Epstein and Koji Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), no. 6, 1275–1289. MR 1452851, DOI 10.1016/S0040-9383(96)00046-8
- Koji Fujiwara, The second bounded cohomology of a group acting on a Gromov-hyperbolic space, Proc. London Math. Soc. (3) 76 (1998), no. 1, 70–94. MR 1476898, DOI 10.1112/S0024611598000033
- Koji Fujiwara, The second bounded cohomology of an amalgamated free product of groups, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1113–1129. MR 1491864, DOI 10.1090/S0002-9947-99-02282-5
- R. I. Grigorchuk, Some results on bounded cohomology, Combinatorial and geometric group theory (Edinburgh, 1993) London Math. Soc. Lecture Note Ser., vol. 204, Cambridge Univ. Press, Cambridge, 1995, pp. 111–163. MR 1320279
- Joel Ryan Louwsma, Extremality of the Rotation Quasimorphism on the Modular Group, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–California Institute of Technology. MR 3047198
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3
- A. H. Rhemtulla, A problem of bounded expressibility in free products, Proc. Cambridge Philos. Soc. 64 (1968), 573–584. MR 225889, DOI 10.1017/s0305004100043231
- W. A. Stein et al., Sage Mathematics Software (Version 5.11), The Sage Development Team, 2013, http://www.sagemath.org.
Additional Information
- Matt Clay
- Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
- Email: mattclay@uark.edu
- Max Forester
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- Email: forester@math.ou.edu
- Joel Louwsma
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- Address at time of publication: Department of Mathematics, Niagara University, P.O. Box 2044, Niagara University, New York 14109
- Email: jlouwsma@gmail.com, jlouwsma@niagara.edu
- Received by editor(s): November 13, 2013
- Received by editor(s) in revised form: May 19, 2014
- Published electronically: September 15, 2015
- Additional Notes: The first author was partially supported by NSF grant DMS-1006898
The second author was partially supported by NSF grant DMS-1105765 - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4751-4785
- MSC (2010): Primary 20F65; Secondary 20E08, 20F12, 57M07
- DOI: https://doi.org/10.1090/tran/6510
- MathSciNet review: 3456160