## Algebraic & Geometric Topology

### Scl in free products

Lvzhou Chen

#### Abstract

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

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

Dates
Revised: 1 May 2018
Accepted: 12 July 2018
First available in Project Euclid: 27 October 2018

https://projecteuclid.org/euclid.agt/1540605643

Digital Object Identifier
doi:10.2140/agt.2018.18.3279

Mathematical Reviews number (MathSciNet)
MR3868221

Zentralblatt MATH identifier
06990064

#### Citation

Chen, Lvzhou. Scl in free products. Algebr. Geom. Topol. 18 (2018), no. 6, 3279--3313. doi:10.2140/agt.2018.18.3279. https://projecteuclid.org/euclid.agt/1540605643

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