Inventiones mathematicae

, Volume 143, Issue 3, pp 523–570

Iteration of mapping classes and limits of hyperbolic 3-manifolds

  • Jeffrey F. Brock

DOI: 10.1007/PL00005799

Cite this article as:
Brock, J. Invent. math. (2001) 143: 523. doi:10.1007/PL00005799


Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {QiX,Y)}i=1 of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface DϕS so that the geometric limits have homeomorphism type S×ℝ-Dϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and Dϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {QsiX,Y)}i=1 converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Jeffrey F. Brock
    • 1
  1. 1.Department of Mathematics, Stanford University, Stanford, CA 94305, USAUS