Algebraic & Geometric Topology

The bumping set and the characteristic submanifold

Genevieve S Walsh

Full-text: Open access


We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group Γ is the boundary of the characteristic submanifold of the associated 3–manifold with boundary. Some examples of interesting characteristic submanifolds are given. We also give a construction of the characteristic submanifold directly from the Nielsen core of the bumping set. The proofs are from “first principles”, using properties of uniform domains and the fact that quasi-conformal discs are uniform domains.

Article information

Algebr. Geom. Topol., Volume 14, Number 1 (2014), 283-297.

Received: 10 August 2012
Revised: 31 May 2013
Accepted: 26 June 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10] 57M60: Group actions in low dimensions

Kleinian group characteristic submanifold


Walsh, Genevieve S. The bumping set and the characteristic submanifold. Algebr. Geom. Topol. 14 (2014), no. 1, 283--297. doi:10.2140/agt.2014.14.283.

Export citation


  • J W Anderson, Intersections of analytically and geometrically finite subgroups of Kleinian groups, Trans. Amer. Math. Soc. 343 (1994) 87–98
  • J W Cannon, C D Feustel, Essential embeddings of annuli and Möbius bands in $3$–manifolds, Trans. Amer. Math. Soc. 215 (1976) 219–239
  • W Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, from: “Geometric topology”, (J C Cantrell, editor), Academic Press, New York (1979) 91–99
  • K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer, Berlin (1979)
  • M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser, Boston, MA (2001)
  • C Lecuire, Plissage des variétés hyperboliques de dimension $3$, Invent. Math. 164 (2006) 85–141
  • O Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics 109, Springer, New York (1987)
  • A Marden, Outer circles: An introduction to hyperbolic $3$–manifolds, Cambridge Univ. Press (2007)
  • B Maskit, Intersections of component subgroups of Kleinian groups, from: “Discontinuous groups and Riemann surfaces”, (L Greenberg, editor), Ann. of Math. Studies 79, Princeton Univ. Press (1974) 349–367
  • C McMullen, Kleinian groups Available at \setbox0\makeatletter\@url {\unhbox0
  • W F Osgood, E H Taylor, Conformal transformations on the boundaries of their regions of definitions, Trans. Amer. Math. Soc. 14 (1913) 277–298
  • W P Thurston, Hyperbolic structures on $3$–manifolds, III: Deformations of $3$–manifolds with incompressible boundary