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Scl in free products

Lvzhou Chen

Algebraic & Geometric Topology 18 (2018) 3279–3313

We study stable commutator length (scl) in free products via surface maps into a wedge of spaces. We prove that scl is piecewise rational linear if it vanishes on each factor of the free product, generalizing a theorem of Danny Calegari. We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show independently and in a new approach that scl in free products of cyclic groups behaves in a piecewise quasirational way when the word is fixed but the orders of factors vary, previously proved by Timothy Susse, settling a conjecture of Alden Walker.

stable commutator length, free product
Mathematical Subject Classification 2010
Primary: 57M07
Secondary: 20E06, 20F12, 20F65, 20J06, 52C07
Received: 21 July 2017
Revised: 1 May 2018
Accepted: 12 July 2018
Published: 18 October 2018
Lvzhou Chen
Department of Mathematics
University of Chicago
Chicago, IL
United States