Abstract
We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for Out(Fn). Then, in joint work with Radhika Gupta, we show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is a direct analog for the cyclic splitting complex of Handel and Mosher's theorem on loxodromics for the free splitting complex. As a step towards proving that all of the loxodromics for this complex are WPD elements, we show that such outer automorphisms have virtually cyclic centralizers.
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