TY - JOUR

T1 - The free splitting complex of a free group, II

T2 - Loxodromic outer automorphisms

AU - Handel, Michael

AU - Mosher, Lee

N1 - Funding Information: Received by the editors June 9, 2017, and, in revised form, July 23, 2018. 2010 Mathematics Subject Classification. Primary 20F65, 57M07; Secondary 20F28, 20E05. The first author was supported by NSF grant DMS-1308710 and various PSC-CUNY grants. The second author was supported by NSF grant DMS-1406376.

PY - 2019/9/15

Y1 - 2019/9/15

N2 - We study the loxodromic elements for the action of Out(Fn) on the free splitting complex of the rank n free group Fn. 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 Out(Fn) 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 Out(Fn) 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.

AB - We study the loxodromic elements for the action of Out(Fn) on the free splitting complex of the rank n free group Fn. 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 Out(Fn) 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 Out(Fn) 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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U2 - https://doi.org/10.1090/tran7698

DO - https://doi.org/10.1090/tran7698

M3 - Article

VL - 372

SP - 4053

EP - 4105

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -