## The free splitting complex of a free group, II: Loxodromic outer automorphisms

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- by Michael Handel and Lee Mosher PDF
- Trans. Amer. Math. Soc.
**372**(2019), 4053-4105 Request permission

## Abstract:

We study the loxodromic elements for the action of $\mathsf {Out}(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. 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 $\mathsf {Out}(F_n)$ 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 $\mathsf {Out}(F_n)$ 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.## References

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## Additional Information

**Michael Handel**- Affiliation: Department of Mathematics, The City University of New York, New York, New York 10016
- MR Author ID: 223960
- Email: michael.handel@lehman.cuny.edu
**Lee Mosher**- Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
- MR Author ID: 248017
- Email: mosher@rutgers.edu
- Received by editor(s): June 9, 2017
- Received by editor(s) in revised form: July 23, 2018
- Published electronically: February 11, 2019
- Additional Notes: 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 4053-4105 - MSC (2010): Primary 20F65, 57M07; Secondary 20F28, 20E05
- DOI: https://doi.org/10.1090/tran/7698
- MathSciNet review: 4009387