Démonstration d’un théorème de Penner sur la composition des twists de Dehn. (Proof of a theorem of Penner on the composition of Dehn twists).

*(French)*Zbl 0779.57005The classification theorem for diffeomorphisms of compact 2-manifolds was developed by Nielsen many years ago, and was completed and integrated into Teichmüller theory by Thurston over a decade ago. Each diffeomorphism either splits into diffeomorphisms of submanifolds, which are again amenable to the classification theorem, or is isotopic to a pseudo-Anosov diffeomorphism. The latter preserves two (singular) measured foliations, upon which it acts in a controlled manner which can be further studied.

One of the simplest known constructions of examples of pseudo-Anosov diffeomorphisms is the following. Let \(\{\gamma_ 1,\ldots,\gamma_ k\}\) and \(\{\delta_ 1,\ldots,\delta_ \ell\}\) be two collections of pairwise disjoint nonparallel noncontractible simple closed curves, whose union fills the surface. This means that for any collections isotopic to them, the complementary regions of their union consist only of discs or annular neighborhoods of boundary components. Then, every composition of positively oriented Dehn twists about the \(\gamma_ i\) and negatively oriented Dehn twists about the \(\delta_ j\) is isotopic to a pseudo- Anosov diffeomorphism, provided that it contains at least one occurrence of a twist about each \(\gamma_ i\) and each \(\delta_ j\). This was proven by R. Penner using train tracks and dual train-tracks.

In this paper, the author gives an alternative proof using measured foliations rather than train tracks. Explicit representatives of the Dehn twists are constructed which act in a simple manner on a singular flat structure associated to the families of curves. Using the horizontal and vertical measured foliations of the structure, the author analyzes the effect of the iterates of the product of the Dehn twists on simple closed curves and arcs in the surface; in particular, no essential arc or simple closed curve is fixed up to isotopy by any iterate, and consequently the product is isotopic to a pseudo-Anosov diffeomorphism. Besides avoiding some of the technical difficulties of train tracks, this approach provides an explicit description of representative diffeomorphisms and their action on the measured foliations. The representative diffeomorphism induces a homeomorphism on a quotient surface, obtained by collapsing finitely many arcs which are contained in leaves of the measured foliations, and this homeomorphism is topologically conjugate to a pseudo-Anosov diffeomorphism.

One of the simplest known constructions of examples of pseudo-Anosov diffeomorphisms is the following. Let \(\{\gamma_ 1,\ldots,\gamma_ k\}\) and \(\{\delta_ 1,\ldots,\delta_ \ell\}\) be two collections of pairwise disjoint nonparallel noncontractible simple closed curves, whose union fills the surface. This means that for any collections isotopic to them, the complementary regions of their union consist only of discs or annular neighborhoods of boundary components. Then, every composition of positively oriented Dehn twists about the \(\gamma_ i\) and negatively oriented Dehn twists about the \(\delta_ j\) is isotopic to a pseudo- Anosov diffeomorphism, provided that it contains at least one occurrence of a twist about each \(\gamma_ i\) and each \(\delta_ j\). This was proven by R. Penner using train tracks and dual train-tracks.

In this paper, the author gives an alternative proof using measured foliations rather than train tracks. Explicit representatives of the Dehn twists are constructed which act in a simple manner on a singular flat structure associated to the families of curves. Using the horizontal and vertical measured foliations of the structure, the author analyzes the effect of the iterates of the product of the Dehn twists on simple closed curves and arcs in the surface; in particular, no essential arc or simple closed curve is fixed up to isotopy by any iterate, and consequently the product is isotopic to a pseudo-Anosov diffeomorphism. Besides avoiding some of the technical difficulties of train tracks, this approach provides an explicit description of representative diffeomorphisms and their action on the measured foliations. The representative diffeomorphism induces a homeomorphism on a quotient surface, obtained by collapsing finitely many arcs which are contained in leaves of the measured foliations, and this homeomorphism is topologically conjugate to a pseudo-Anosov diffeomorphism.

Reviewer: D.McCullough (Norman)

##### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57R30 | Foliations in differential topology; geometric theory |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

57R50 | Differential topological aspects of diffeomorphisms |

30F99 | Riemann surfaces |

##### Keywords:

diffeomorphisms of compact 2-manifolds; examples of pseudo-Anosov diffeomorphisms; collections of pairwise disjoint nonparallel noncontractible simple closed curves; composition of Dehn twists; measured foliations; action on measured foliations##### References:

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