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

Ken’ichi Ohshika

doi:10.5802/afst.1647

Ken’ichi Ohshika ¹

¹ Department of Mathematics, Faculty of Science, Gakushuin University, Tokyo 171-8588 (Japan)

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

Résumé
Abstract

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.

Reçu le : 2015-06-08
Accepté le : 2018-11-28
Publié le : 2020-12-09

DOI : 10.5802/afst.1647

Affiliations des auteurs :

Ken’ichi Ohshika ¹

¹ Department of Mathematics, Faculty of Science, Gakushuin University, Tokyo 171-8588 (Japan)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2020_6_29_4_805_0,
     author = {Ken{\textquoteright}ichi Ohshika},
     title = {Divergence, exotic convergence and self-bumping in {quasi-Fuchsian} spaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {805--895},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {4},
     year = {2020},
     doi = {10.5802/afst.1647},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1647/}
}

TY  - JOUR
AU  - Ken’ichi Ohshika
TI  - Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2020
SP  - 805
EP  - 895
VL  - 29
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1647/
DO  - 10.5802/afst.1647
LA  - en
ID  - AFST_2020_6_29_4_805_0
ER  -

%0 Journal Article
%A Ken’ichi Ohshika
%T Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2020
%P 805-895
%V 29
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1647/
%R 10.5802/afst.1647
%G en
%F AFST_2020_6_29_4_805_0

Ken’ichi Ohshika. Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 805-895. doi : 10.5802/afst.1647. https://afst.centre-mersenne.org/articles/10.5802/afst.1647/

Bibliographie
Cité par

[1] William Abikoff On boundaries of Teichmüller spaces and on Kleinian groups. III, Acta Math., Volume 134 (1975), pp. 211-237 | DOI | MR | Zbl

[2] James W. Anderson; Richard D. Canary Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent. Math., Volume 126 (1996) no. 2, pp. 205-214 | DOI | MR | Zbl

[3] James W. Anderson; Cyril Lecuire Strong convergence of Kleinian groups: the cracked eggshell, Comment. Math. Helv., Volume 88 (2013) no. 4, pp. 813-857 | DOI | MR | Zbl

[4] Lipman Bers On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. Math., Volume 91 (1970), pp. 570-600 | DOI | Zbl

[5] Francis Bonahon Bouts des variétés hyperboliques de dimension $3$ , Ann. Math., Volume 124 (1986) no. 1, pp. 71-158 | DOI | MR | Zbl

[6] Brian H. Bowditch End invariants of hyperbolic 3-manifolds (preprint)

[7] Brian H. Bowditch The ending lamination theorem (preprint)

[8] Brian H. Bowditch Geometric models for hyperbolic 3-manifolds (preprint)

[9] Jeffrey F. Brock Continuity of Thurston’s length function, Geom. Funct. Anal., Volume 10 (2000) no. 4, pp. 741-797 | DOI | MR | Zbl

[10] Jeffrey F. Brock; Kenneth W. Bromberg On the density of geometrically finite Kleinian groups, Acta Math., Volume 192 (2004) no. 1, pp. 33-93 corrigendum in ibid. 219 (2017), n^o 1, p. 17-19 | DOI | MR | Zbl

[11] Jeffrey F. Brock; Kenneth W. Bromberg; Richard D. Canary; Cyril Lecuire Convergence and divergence of Kleinian surface groups, J. Topol., Volume 8 (2015) no. 3, pp. 811-841 | DOI | MR | Zbl

[12] Jeffrey F. Brock; Kenneth W. Bromberg; Richard D. Canary; Yair N. Minsky Local topology in deformation spaces of hyperbolic 3-manifolds, Geom. Topol., Volume 15 (2011) no. 2, pp. 1169-1224 | DOI | MR | Zbl

[13] Jeffrey F. Brock; Richard D. Canary; Yair N. Minsky The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. Math., Volume 176 (2012) no. 1, pp. 1-149 | DOI | MR | Zbl

[14] Kenneth W. Bromberg Hyperbolic Dehn surgery on geometrically infinite 3-manifolds (2000) (https://arxiv.org/abs/math/0009150)

[15] Kenneth W. Bromberg Drilling short geodesics in hyperbolic 3-manifolds, Spaces of Kleinian groups (London Mathematical Society Lecture Note Series), Volume 329, Cambridge University Press, 2006, pp. 1-27 | MR | Zbl

[16] Kenneth W. Bromberg Projective structures with degenerate holonomy and the Bers density conjecture, Ann. Math., Volume 166 (2007) no. 1, pp. 77-93 | DOI | MR | Zbl

[17] Kenneth W. Bromberg The space of Kleinian punctured torus groups is not locally connected, Duke Math. J., Volume 156 (2011), pp. 387-427 | DOI | MR | Zbl

[18] Richard D. Canary A covering theorem for hyperbolic $3$ -manifolds and its applications, Topology, Volume 35 (1996) no. 3, pp. 751-778 | DOI | MR | Zbl

[19] Richard D. Canary Introductory bumponomics: the topology of deformation spaces of hyperbolic 3-manifolds, Teichmüller theory and moduli problem (Ramanujan Mathematical Society Lecture Notes Series), Volume 10, Ramanujan Mathematical Society, 2010, pp. 131-150 | MR | Zbl

[20] Charalampos Charitos; Ioannis Papadoperakis; Athanase Papadopoulos On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface, Proc. Am. Math. Soc., Volume 142 (2014) no. 6, pp. 2179-2191 | DOI | MR | Zbl

[21] Michel Coornaert; Thomas Delzant; Athanase Papadopoulos Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics, 1441, Springer, 1990, x+165 pages | Zbl

[22] David B. A. Epstein; Albert Marden Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) (London Mathematical Society Lecture Note Series), Volume 111, Cambridge University Press, 1987, pp. 113-253 | MR | Zbl

[23] A. Fathi; V. Poénaru; F. Laudenbach Travaux de Thurston sur les surfaces, Astérisque, 66, Société Mathématique de France, 1979, 284 pages (Séminaire Orsay, With an English summary) | Numdam | MR

[24] Michael Freedman; Joel Hass; Peter Scott Least area incompressible surfaces in $3$ -manifolds, Invent. Math., Volume 71 (1983) no. 3, pp. 609-642 | DOI | MR | Zbl

[25] William J. Harvey Boundary structure of the modular group, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Annals of Mathematics Studies), Volume 97, Princeton University Press, 1981, pp. 245-251 | DOI | MR | Zbl

[26] Kentaro Ito Convergence and divergence of Kleinian punctured torus groups, Am. J. Math., Volume 134 (2012) no. 4, pp. 861-889 | MR | Zbl

[27] Troels Jørgensen; Albert Marden Algebraic and geometric convergence of Kleinian groups, Math. Scand., Volume 66 (1990) no. 1, pp. 47-72 | DOI | MR | Zbl

[28] Steven P. Kerckhoff The asymptotic geometry of Teichmüller space, Topology, Volume 19 (1980) no. 1, pp. 23-41 | DOI | MR | Zbl

[29] Steven P. Kerckhoff The Nielsen realization problem, Ann. Math., Volume 117 (1983) no. 2, pp. 235-265 | DOI | MR | Zbl

[30] Steven P. Kerckhoff; William P. Thurston Noncontinuity of the action of the modular group at Bers’ boundary of Teichmüller space, Invent. Math., Volume 100 (1990) no. 1, pp. 25-47 | DOI | Zbl

[31] Erica Klarreich The boundary at infinity of the curve complex and the relative Teichmüller space (2018) (https://arxiv.org/abs/1803.10339)

[32] Gero Kleineidam; Juan Souto Algebraic convergence of function groups, Comment. Math. Helv., Volume 77 (2002) no. 2, pp. 244-269 | DOI | MR | Zbl

[33] Cyril Lecuire An extension of the Masur domain, Spaces of Kleinian groups (London Mathematical Society Lecture Note Series), Volume 329, Cambridge University Press, 2006, pp. 49-73 | DOI | MR | Zbl

[34] Aaaron Magid Deformation spaces of Kleinian surface groups are not locally connected, Geom. Topol., Volume 16 (2012), pp. 1247-1320 | DOI | MR | Zbl

[35] Albert Marden Outer circles. An introduction to hyperbolic 3-manifolds, Cambridge University Press, 2007, xviii+427 pages | Zbl

[36] Albert Marden Hyperbolic manifolds. An introduction in 2 and 3 dimensions, Cambridge University Press, 2016, xviii+515 pages | Zbl

[37] Howard A. Masur; Yair N. Minsky Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., Volume 138 (1999) no. 1, pp. 103-149 | DOI | MR | Zbl

[38] Howard A. Masur; Yair N. Minsky Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal., Volume 10 (2000) no. 4, pp. 902-974 | DOI | MR | Zbl

[39] Curtis T. McMullen Complex earthquakes and Teichmüller theory, J. Am. Math. Soc., Volume 11 (1998) no. 2, pp. 283-320 | DOI | MR | Zbl

[40] Yair N. Minsky The classification of Kleinian surface groups. I. Models and bounds, Ann. Math., Volume 171 (2010) no. 1, pp. 1-107 | DOI | MR | Zbl

[41] Hossein Namazi; Juan Souto Non-realizability and ending laminations: proof of the density conjecture, Acta Math., Volume 209 (2012) no. 2, pp. 323-395 | DOI | MR | Zbl

[42] Ken’ichi Ohshika Ending laminations and boundaries for deformation spaces of Kleinian groups, J. Lond. Math. Soc., Volume 42 (1990) no. 1, pp. 111-121 | DOI | MR | Zbl

[43] Ken’ichi Ohshika Limits of geometrically tame Kleinian groups, Invent. Math., Volume 99 (1990) no. 1, pp. 185-203 | DOI | MR | Zbl

[44] Ken’ichi Ohshika Geometric behaviour of Kleinian groups on boundaries for deformation spaces, Q. J. Math., Oxf. II. Ser., Volume 43 (1992) no. 169, pp. 97-111 | DOI | MR | Zbl

[45] Ken’ichi Ohshika Divergent sequences of Kleinian groups, The Epstein birthday schrift (Geometry and Topology Monographs), Volume 1, Geometry and Topology Publications, 1998, pp. 419-450 | DOI | MR | Zbl

[46] Ken’ichi Ohshika Realising end invariants by limits of minimally parabolic, geometrically finite groups, Geom. Topol., Volume 15 (2011) no. 2, pp. 827-890 | DOI | MR | Zbl

[47] Ken’ichi Ohshika A note on the rigidity of unmeasured lamination spaces, Proc. Am. Math. Soc., Volume 141 (2013) no. 12, pp. 4385-4389 | DOI | MR | Zbl

[48] Ken’ichi Ohshika Compactifications of Teichmüller spaces, Handbook of Teichmüller theory. Vol. IV (IRMA Lectures in Mathematics and Theoretical Physics), Volume 19, European Mathematical Society, 2014, pp. 235-254 | MR | Zbl

[49] Ken’ichi Ohshika Reduced Bers boundaries of Teichmüller spaces, Ann. Inst. Fourier, Volume 64 (2014) no. 1, pp. 145-176 | DOI | Numdam | Zbl

[50] Ken’ichi Ohshika; Teruhiko Soma Geometry and topology of geometric limits (2010) (https://arxiv.org/abs/1002.4266v1)

[51] Athanase Papadopoulos A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface, Proc. Am. Math. Soc., Volume 136 (2008) no. 12, pp. 4453-4460 | DOI | MR | Zbl

[52] Athanase Papadopoulos Actions of mapping class groups, Handbook of group actions. Vol. I (Advanced Lectures in Mathematics (ALM)), Volume 31, International Press, 2015, pp. 189-248 | MR

[53] Dennis Sullivan Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups, Acta Math., Volume 155 (1985) no. 3-4, pp. 243-260 | DOI | MR | Zbl

[54] William P. Thurston The geometry and topology of $3$ -manifolds (Princeton Univ. Lecture Notes)

[55] William P. Thurston Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle (1998) (https://arxiv.org/abs/math/9801045)

[56] William P. Thurston Minimal stretch maps between hyperbolic surfaces (1998) (https://arxiv.org/abs/math/9801039)

[57] Friedhelm Waldhausen On irreducible $3$ -manifolds which are sufficiently large, Ann. Math., Volume 87 (1968), pp. 56-88 | DOI | MR | Zbl

Cité par Sources :