## 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

### Keywords:

measured lamination; $${\mathbb{R}}$$-tree; train track
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### References:

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