Abstract
In [4] a space X n was introduced on which the group Out(F n ) of outer automorphisms of a free group of rank n acts virtually freely. Since then, this space has come to be known as “outer space.” Outer space can be defined as a space of free actions of F n on simplicial ℝ-trees; we require that all actions be minimal, and we identify two actions if they differ only by scaling the metric on the ℝ-tree. To describe the topology on outer space, we associate to each action α: F n × T → T a length function | · | α: F n → ℝ defined by
where d is the distance in the tree T. We have | g | α = | h −1 gh| α and | · |α ≡ 0 if and only if some point of T is fixed by all of F n . Thus an action with no fixed point determines a point in ℝc — {0}, where C is the set of conjugacy classes in F n . Since actions differing by a scalar multiple define the same point of outer space, we have a map from X n to the infinite dimensional projective space P c = ℝc — {0}/ℝ*. It can be shown that this map is injective (see [3] or [1]). We topologize X n as a subspace of P c.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
R. Alperin and H. Bass, Length Functions of Group Actions on A- trees, in “Combinatorial Group Theory and Topology,” S.M. Gersten et al., eds., Princeton University Press, Princeton, 1987, pp. 265–378.
M. Cohen and M. Lustig, Very Small Group Actions on R-trees and Dehn-twist Automorphisms.
M. Culler and J.W. Morgan, Group Actions on R-trees, Proc. Lond. Math. Soc. 55 (1987), 571–604.
M. Culler and K. Vogtmann, Moduli of Graphs and Automorphisms of Free Groups, Invent. Math. 84 (1986), 91–119.
H. Gillet and P.B. Shalen, Dendrology of Groups of Low Q-ranks, preprint, J. Diff. Geom. (1988) (to appear).
A.E. Hatcher, Measured Lamination Spaces for Surfaces, from the Topological Viewpoint, Topology and its Applications 30 (1988), 63–88.
J.W. Morgan, Ergodic Theory and Free Actions on Trees, Invent. Math. 94 (1988), 605–622.
J.W. Morgan and J.P. Otal, Relative Growth Rates of Closed Geodesics on a Surface under Varying Hyperbolic Structures, Comment. Math. Helv. (to appear).
J.W. Morgan and P.B. Shalen, Degenerations of Hyperbolic Structures, II: Measured Laminations in 3-Manifolds, Ann. of Math. 127 (1988), 403–456.
W. Parry. these proceedings.
F. Paulin, Topologie de Gromov equivariante, Structures Hyperboliques et Arbres Reels, Invent. Math. 94 (1988), 53–80.
J. Plante, Foliations with Measure-Preserving Holonomy, Ann. of Math. 102 (1975), 327–361.
J.P. Serre, “Trees,” Springer-Verlag, Berlin, 1980. Translation of “Arbres, Amalgames, SL2i” Astérisque 46, 1977.
R. Skora, Splittings of surfaces, preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Culler, M., Vogtmann, K. (1991). The Boundary of Outer Space in Rank Two. In: Alperin, R.C. (eds) Arboreal Group Theory. Mathematical Sciences Research Institute Publications, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3142-4_8
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3142-4_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7811-5
Online ISBN: 978-1-4612-3142-4
eBook Packages: Springer Book Archive