×

Splittings of surfaces. (English) Zbl 0708.30044

The author investigates the action on an \({\mathbb{R}}\)-tree of the fundamental group \(\pi_ 1(F)\) of a closed surface F of negative Euler characteristic. The main result is that such an action \(\pi_ 1(F)\times T\to T\) corresponds to a measured geodesic lamination in F if and only if the stabilizer of each arc in T is cyclic; this extends the work of various authors who have approached this result under additional hypotheses. This implies that the space of projective measured geodesic laminations on F forms a natural boundary of the space of discrete faithful representations of \(\pi_ 1(F)\) into the isometries of hyperbolic n-space for each \(n\geq 2\); this extends the famous work of Thurston in case \(n=2,3\). Further applications of the main result are also discussed. The tools required for the proofs include train tracks and interval exchange transformations.
Reviewer: R.Penner

MSC:

30F25 Ideal boundary theory for Riemann surfaces
22E40 Discrete subgroups of Lie groups
30F60 Teichmüller theory for Riemann surfaces
20F65 Geometric group theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Marc Culler and John W. Morgan, Group actions on \?-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571 – 604. · Zbl 0658.20021
[2] Henri Gillet and Peter B. Shalen, Dendrology of groups in low \?-ranks, J. Differential Geom. 32 (1990), no. 3, 605 – 712. · Zbl 0732.20011
[3] A. E. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30 (1988), no. 1, 63 – 88. · Zbl 0662.57005
[4] John W. Morgan, Group actions on trees and the compactification of the space of classes of \?\?(\?,1)-representations, Topology 25 (1986), no. 1, 1 – 33. · Zbl 0595.57030
[5] John W. Morgan and Jean-Pierre Otal, Relative growth rates of closed geodesics on a surface under varying hyperbolic structures, Comment. Math. Helv. 68 (1993), no. 2, 171 – 208. · Zbl 0795.57009
[6] John W. Morgan and Peter B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2) 120 (1984), no. 3, 401 – 476. · Zbl 0583.57005
[7] J. W. Morgan and P. B. Shalen, Surface groups acting on R-trees, preprint.
[8] R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. · Zbl 0765.57001
[9] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. · Zbl 0548.20018
[10] S. M. Gersten , Essays in group theory, Mathematical Sciences Research Institute Publications, vol. 8, Springer-Verlag, New York, 1987. · Zbl 0626.00014
[11] R. K. Skora, Geometric actions of surface groups on \Lambda -trees, Comm. Math. (to appear). · Zbl 0724.57008
[12] W. P. Thurston, Geometry and topology of 3-manifolds, preprint.
[13] W. P. Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.