Geometry & Topology

Convergence properties of end invariants

Jeffrey F Brock, Kenneth W Bromberg, Richard D Canary, and Yair N Minsky

Full-text: Open access

Abstract

We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.

Article information

Source
Geom. Topol., Volume 17, Number 5 (2013), 2877-2922.

Dates
Received: 23 August 2012
Revised: 16 January 2013
Accepted: 19 March 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732690

Digital Object Identifier
doi:10.2140/gt.2013.17.2877

Mathematical Reviews number (MathSciNet)
MR3190301

Zentralblatt MATH identifier
06217242

Subjects
Primary: 30F40: Kleinian groups [See also 20H10] 57M50: Geometric structures on low-dimensional manifolds

Keywords
Kleinian group hyperbolic 3–manifold end invariant ending lamination

Citation

Brock, Jeffrey F; Bromberg, Kenneth W; Canary, Richard D; Minsky, Yair N. Convergence properties of end invariants. Geom. Topol. 17 (2013), no. 5, 2877--2922. doi:10.2140/gt.2013.17.2877. https://projecteuclid.org/euclid.gt/1513732690


Export citation

References

  • I Agol, Tameness of hyperbolic $3$–manifolds
  • J W Anderson, R D Canary, Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent. Math. 126 (1996) 205–214
  • J W Anderson, R D Canary, Cores of hyperbolic $3$–manifolds and limits of Kleinian groups, Amer. J. Math. 118 (1996) 745–779
  • L Bers, An inequality for Riemann surfaces, from: “Differential geometry and complex analysis”, (C I, H M Farkas, editors), Springer, Berlin (1985) 87–93
  • F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71–158
  • B H Bowditch, Length bounds on curves arising from tight geodesics, Geom. Funct. Anal. 17 (2007) 1001–1042
  • J Brock, Continuity of Thurston's length function, Geom. Funct. Anal. 10 (2000) 741–797
  • J Brock, Boundaries of Teichmüller spaces and end-invariants for hyperbolic $3$–manifolds, Duke Math. J. 106 (2001) 527–552
  • J Brock, K Bromberg, R D Canary, C Lecuire, Convergence and divergence of Kleinian surface groups, in preparation
  • J Brock, K W Bromberg, R D Canary, Y N Minsky, Local topology in deformation spaces of hyperbolic $3$–manifolds, Geom. Topol. 15 (2011) 1169–1224
  • J Brock, R D Canary, Y N Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. 176 (2012) 1–149
  • K Bromberg, The space of Kleinian punctured torus groups is not locally connected, Duke Math. J. 156 (2011) 387–427
  • D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 385–446
  • R D Canary, A covering theorem for hyperbolic $3$–manifolds and its applications, Topology 35 (1996) 751–778
  • R D Canary, Introductory bumponomics: The topology of deformation spaces of hyperbolic $3$–manifolds, from: “Teichmüller theory and moduli problem”, (I Biswas, R S Kulkarni, S Mitra, editors), Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Math. Soc., Mysore (2010) 131–150
  • D B A Epstein, A Marden, V Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. 159 (2004) 305–336
  • U Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, from: “Spaces of Kleinian groups”, (Y N Minsky, M Sakuma, C Series, editors), London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press (2006) 187–207
  • E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space Available at \setbox0\makeatletter\@url http://ericaklarreich.com/curvecomplex.pdf {\unhbox0
  • C J Leininger, S Schleimer, Connectivity of the space of ending laminations, Duke Math. J. 150 (2009) 533–575
  • A D Magid, Deformation spaces of Kleinian surface groups are not locally connected, Geom. Topol. 16 (2012) 1247–1320
  • H A Masur, Y N Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • H A Masur, Y N Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902–974
  • C T McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998) 283–320
  • Y N Minsky, The classification of punctured-torus groups, Ann. of Math. 149 (1999) 559–626
  • Y N Minsky, Kleinian groups and the complex of curves, Geom. Topol. 4 (2000) 117–148
  • Y N Minsky, The classification of Kleinian surface groups, I: Models and bounds, Ann. of Math. 171 (2010) 1–107
  • K Ohshika, Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces Available at \setbox0\makeatletter\@url http://front.math.ucdavis.edu/1010.0070 {\unhbox0
  • W P Thurston, The geometry and topology of $3$–manifolds, lecture notes, Princeton University (1978–1981) Available at \setbox0\makeatletter\@url http://library.msri.org/books/gt3m {\unhbox0