Algebraic & Geometric Topology

Scl in free products

Lvzhou Chen

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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.

Article information

Algebr. Geom. Topol., Volume 18, Number 6 (2018), 3279-3313.

Received: 21 July 2017
Revised: 1 May 2018
Accepted: 12 July 2018
First available in Project Euclid: 27 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M07: Topological methods in group theory
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20F12: Commutator calculus 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20J06: Cohomology of groups 52C07: Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]

stable commutator length free product


Chen, Lvzhou. Scl in free products. Algebr. Geom. Topol. 18 (2018), no. 6, 3279--3313. doi:10.2140/agt.2018.18.3279.

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