Calegari, Daniel (2011) Scl, sails, and surgery. Journal of Topology, 4 (2). pp. 305-326. ISSN 1753-8416 http://resolver.caltech.edu/CaltechAUTHORS:20110726-095541078
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We establish a close connection between the stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programing. Furthermore, we show that the scl spectrum of non-Abelian free groups contains elements congruent to every rational number modulo ℤ, and contains well-ordered sequences of values with ordinal type ω^ω. Finally, we study families of elements w(p) in free groups obtained by surgery on a fixed element w in a free product of Abelian groups of higher rank, and show that scl(w(p)) → scl(w) as p → ∞.
|Additional Information:||© 2011 London Mathematical Society. Received November 10, 2009; revision received November 3, 2010. Published online 25 March 2011. Danny Calegari was partially supported by NSF grants DMS 0707130 and DMS 1005246. It is a pleasure to acknowledge my own great intellectual debt to John, and it seems especially serendipitous to discover, in relatively unheralded work he did in the later part of his life, some beautiful new ideas which continue to inform and inspire. I would like to thank Lukas Brantner, Jon McCammond, Alden Walker, and the referee for helpful comments and corrections.|
|Classification Code:||2000 Mathematics Subject Classifiication 57M07, 20F65, 20J05, 20J06, 20F12, 11J70|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Jason Perez|
|Deposited On:||26 Jul 2011 20:17|
|Last Modified:||26 Jul 2011 20:17|
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