Virtually abelian subgroups of IAₙ(Z/3) are abelian

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When studying subgroups of Out(Fₙ), one often replaces a given subgroup H with one of its finite index subgroups H₀ so that virtual properties of H become actual properties of H₀. In many cases, the finite index subgroup is H₀ = H ∩ IAₙ(Z/3). For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of IAₙ(Z/3) is abelian.