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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Parageometric outer automorphisms of free groups
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by Michael Handel and Lee Mosher PDF
Trans. Amer. Math. Soc. 359 (2007), 3153-3183 Request permission

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
  • MR Author ID: 223960
  • Email: michael.handel@lehman.cuny.edu
  • Lee Mosher
  • Affiliation: Department of Mathematics and Computer Science, Rutgers University at Newark, Newark, New Jersey 07102
  • MR Author ID: 248017
  • Email: mosher@andromeda.rutgers.edu
  • 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.
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3153-3183
  • MSC (2000): Primary 20E05; Secondary 20E36, 20F65
  • DOI: https://doi.org/10.1090/S0002-9947-07-04065-2
  • MathSciNet review: 2299450