Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Michael Handel and Lee Mosher
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
Published electronically: February 11, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F65, 57M07, 20F28, 20E05

Retrieve articles in all journals with MSC (2010): 20F65, 57M07, 20F28, 20E05


Additional Information

Michael Handel
Affiliation: Department of Mathematics, The City University of New York, New York, New York 10016
Email: michael.handel@lehman.cuny.edu

Lee Mosher
Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
Email: mosher@rutgers.edu

DOI: https://doi.org/10.1090/tran/7698
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.
Article copyright: © Copyright 2019 American Mathematical Society