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**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 |

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\textit{R. K. Skora}, Bull. Am. Math. Soc., New Ser. 23, No. 1, 85--90 (1990; Zbl 0708.30044)

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### References:

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