## Parageometric outer automorphisms of free groups

HTML articles powered by AMS MathViewer

- 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.

## References

- M. Bestvina and M. Feighn,
*Outer limits*, preprint. - Mladen Bestvina and Mark Feighn,
*Stable actions of groups on real trees*, Invent. Math.**121**(1995), no. 2, 287–321. MR**1346208**, DOI 10.1007/BF01884300 - M. Bestvina, M. Feighn, and M. Handel,
*Laminations, trees, and irreducible automorphisms of free groups*, Geom. Funct. Anal.**7**(1997), no. 2, 215–244. MR**1445386**, DOI 10.1007/PL00001618 - Mladen Bestvina, Mark Feighn, and Michael Handel,
*The Tits alternative for $\textrm {Out}(F_n)$. I. Dynamics of exponentially-growing automorphisms*, Ann. of Math. (2)**151**(2000), no. 2, 517–623. MR**1765705**, DOI 10.2307/121043 - Mladen Bestvina, Mark Feighn, and Michael Handel,
*The Tits alternative for $\textrm {Out}(F_n)$. II. A Kolchin type theorem*, Ann. of Math. (2)**161**(2005), no. 1, 1–59. MR**2150382**, DOI 10.4007/annals.2005.161.1 - Mladen Bestvina and Michael Handel,
*Train tracks and automorphisms of free groups*, Ann. of Math. (2)**135**(1992), no. 1, 1–51. MR**1147956**, DOI 10.2307/2946562 - Marshall M. Cohen and Martin Lustig,
*Very small group actions on $\textbf {R}$-trees and Dehn twist automorphisms*, Topology**34**(1995), no. 3, 575–617. MR**1341810**, DOI 10.1016/0040-9383(94)00038-M - Marc Culler and John W. Morgan,
*Group actions on $\textbf {R}$-trees*, Proc. London Math. Soc. (3)**55**(1987), no. 3, 571–604. MR**907233**, DOI 10.1112/plms/s3-55.3.571 - Marc Culler and Karen Vogtmann,
*Moduli of graphs and automorphisms of free groups*, Invent. Math.**84**(1986), no. 1, 91–119. MR**830040**, DOI 10.1007/BF01388734 *Travaux de Thurston sur les surfaces*, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR**568308**- Damien Gaboriau, Andre Jaeger, Gilbert Levitt, and Martin Lustig,
*An index for counting fixed points of automorphisms of free groups*, Duke Math. J.**93**(1998), no. 3, 425–452. MR**1626723**, DOI 10.1215/S0012-7094-98-09314-0 - Vincent Guirardel,
*Cœur et nombre d’intersection pour les actions de groupes sur les arbres*, Ann. Sci. École Norm. Sup. (4)**38**(2005), no. 6, 847–888 (French, with English and French summaries). MR**2216833**, DOI 10.1016/j.ansens.2005.11.001 - M. Handel and L. Mosher,
*The expansion factors of an outer automorphism and its inverse*, Trans. Amer. Math. Soc., this issue. - M. Handel and L. Mosher,
*Axes in outer space*, Preprint, arXiv:math.GR/0605355, 2006. - Gilbert Levitt and Martin Lustig,
*Irreducible automorphisms of $F_n$ have north-south dynamics on compactified outer space*, J. Inst. Math. Jussieu**2**(2003), no. 1, 59–72. MR**1955207**, DOI 10.1017/S1474748003000033 - Douglas Lind and Brian Marcus,
*An introduction to symbolic dynamics and coding*, Cambridge University Press, Cambridge, 1995. MR**1369092**, DOI 10.1017/CBO9780511626302 - Gilbert Levitt and Frédéric Paulin,
*Geometric group actions on trees*, Amer. J. Math.**119**(1997), no. 1, 83–102. MR**1428059**, DOI 10.1353/ajm.1997.0003 - John McCarthy,
*A “Tits-alternative” for subgroups of surface mapping class groups*, Trans. Amer. Math. Soc.**291**(1985), no. 2, 583–612. MR**800253**, DOI 10.1090/S0002-9947-1985-0800253-8 - Karen Vogtmann,
*Automorphisms of free groups and outer space*, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 1–31. MR**1950871**, DOI 10.1023/A:1020973910646

## 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