Abstract
We study the loxodromic elements for the action of O u t ( F n ) \mathsf {Out}(F_n) on the free splitting complex of the rank n n free group F n F_n . Each outer automorphism is either loxodromic or has bounded orbits without any periodic point, or has a periodic point; all three possibilities can occur. Two loxodromic elements are either coaxial or independent, meaning that their attracting-repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each alternative is characterized in terms of attracting laminations; in particular, an outer automorphism is loxodromic if and only if it has a filling attracting lamination. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study, we describe the structure of the subgroup of O u t ( F n ) \mathsf {Out}(F_n) that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of O u t ( F n ) \mathsf {Out}(F_n) on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the weak proper discontinuity (WPD) property of Bestvina and Fujiwara.
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