Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces

Ohshika, Ken’ichi

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 805-895.

Dans cet article, on étudie la topologie des bords des espaces quasi-fuchsiens. D’abord on montre comment on peut savoir les invariants des bouts du groupe limite pour une suite convergente de groupes quasi-fuchsiens donnée, en utilisant les informations sur le comportement asymptotique des structures conformes à l’infini des groupes dans la suite. Ce résultat donne lieu à une condition suffisante pour la divergence des groupes quasi-fuchsiens, laquelle est une généralisation du résultat d’Ito qui n’a traité que le cas des groupes du tore une fois perforé. On démontre de plus que des groupes quasi-fuchsiens ne peuvent approcher un b-groupe hors de la tranche de Bers que si la limite admet un locus parabolique isolé. Ce résultat-ci permet également de donner une condition nécessaire pour qu’un point au bord de l’espace de déformations soit un point de « l’entrechoquement ». Pour démontrer ces résultats, on utilise des variétés modèles construites par Minsky et leurs limites géométriques étudiées par Ohshika–Soma. Pour que les lecteurs n’aient pas besoin de se reporter à l’article d’Ohshika–Soma, le présent article aussi contient les arguments simplifiés mais assez détaillés d’Ohshika–Soma qui sont nécessaires pour les démonstrations des théorèmes principaux.

In this paper, we study the topology of the boundaries of quasi-Fuchsian spaces. We first show for a given convergent sequence of quasi-Fuchsian groups, how we can know the end invariant of the limit group from the information on the behaviour of conformal structures at infinity of the groups. This result gives rise to a sufficient condition for divergence of quasi-Fuchsian groups, which generalises Ito’s result in the once-punctured torus case to higher genera. We further show that quasi-Fuchsian groups can approach a b-group not along Bers slices only when the limit has isolated parabolic loci. This makes it possible to give a necessary condition for points on the boundaries of quasi-Fuchsian spaces to be self-bumping points. We use model manifolds invented by Minsky and their geometric limits studied by Ohshika–Soma to prove these results. This paper has been made as self-contained as possible so that the reader does not need to consult the paper of Ohshika–Soma directly.

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