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


30F25 Ideal boundary theory for Riemann surfaces
22E40 Discrete subgroups of Lie groups
30F60 Teichm├╝ller theory for Riemann surfaces
20F65 Geometric group theory
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