2014/02/08 by Michael Handel, Lee Mosher, Handel, Michael +1
Mathematics · #20F65 57M07 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1402.1886
openalex publication_date 2014/02/08 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We study the loxodromic elements for the action of Out(Fn) on the free\nsplitting complex of the rank n free group Fn. We prove that each outer\nautomorphism is either loxodromic, or has bounded orbits without any periodic\npoint, or has a periodic point; and we prove that all three possibilities can\noccur. We also prove that two loxodromic elements are either co-axial or\nindependent, meaning that their attracting/repelling fixed point pairs on the\nGromov boundary of the free splitting complex are either equal or disjoint as\nsets. Each of the alternatives in these results is also characterized in terms\nof the attracting/repelling lamination pairs of an outer automorphism. As an\napplication, each attracting lamination determines its corresponding repelling\nlamination independent of the outer automorphism. As part of this study we\ndescribe the structure of the subgroup of Out(Fn) that stabilizes the fixed\npoint pair of a given loxodromic outer automorphism, and we give examples which\nshow that this subgroup need not be virtually cyclic. As an application, the\naction of Out(Fn) on the free splitting complex is not acylindrical, and its\nloxodromic elements do not all satisfy the WPD property of Bestvina and\nFujiwara.\n