Scl in free products

TL;DR

This paper investigates stable commutator length (scl) in free products, establishing piecewise rational linearity under certain conditions, and introduces new methods for computing scl, including a proof of a conjecture about its behavior in free products of cyclic groups.

Contribution

It generalizes previous results on scl by proving piecewise rational linearity when scl vanishes on factors and introduces a new approach for computing scl in free products.

Findings

scl is piecewise rational linear when it vanishes on each factor

isometric embedding with respect to scl is preserved under free products

scl in free products of cyclic groups varies in a piecewise quasi-rational manner

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 the main result in Danny Calegari's paper "Scl, sails and surgery". 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 quasi-rational way when the word is fixed but the orders of factors vary, previously proved by Timothy Susse, settling a conjecture of Alden Walker.