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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Parageometric outer automorphisms of free groups


Authors: Michael Handel and Lee Mosher
Journal: Trans. Amer. Math. Soc. 359 (2007), 3153-3183
MSC (2000): Primary 20E05; Secondary 20E36, 20F65
Published electronically: February 8, 2007
MathSciNet review: 2299450
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Abstract: We study those fully irreducible outer automorphisms $ \phi$ of a finite rank free group $ F_r$ which are parageometric, meaning that the attracting fixed point of $ \phi$ in the boundary of outer space is a geometric $ \mathbf{R}$-tree with respect to the action of $ F_r$, but $ \phi$ itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion factor of $ \phi$ is strictly larger than the expansion factor of $ \phi^{-1}$. As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism $ \phi$ is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric $ \mathbf{R}$-trees.


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Additional Information

Michael Handel
Affiliation: Department of Mathematics and Computer Science, Lehman College - CUNY, 250 Bedford Park Boulevard W, Bronx, New York 10468
Email: michael.handel@lehman.cuny.edu

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

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04065-2
PII: S 0002-9947(07)04065-2
Received by editor(s): December 9, 2004
Received by editor(s) in revised form: April 22, 2005
Published electronically: February 8, 2007
Additional Notes: The first author was supported in part by NSF grant DMS0103435.
The second author was supported in part by NSF grant DMS0103208.
Article copyright: © Copyright 2007 American Mathematical Society