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Stable commutator length in Baumslag-Solitar groups and quasimorphisms for tree actions


Authors: Matt Clay, Max Forester and Joel Louwsma
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
Published electronically: September 15, 2015
MathSciNet review: 3456160
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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)$.


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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

DOI: https://doi.org/10.1090/tran/6510
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
Article copyright: © Copyright 2015 American Mathematical Society