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.
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