, Volume 143, Issue 3, pp 523-570

Iteration of mapping classes and limits of hyperbolic 3-manifolds

Abstract.

Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q i X,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 D ϕS so that the geometric limits have homeomorphism type S×ℝ-D ϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and D ϕ 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 {Q si X,Y)} i=1 converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.

Oblatum 4-I-1999 & 19-VII-2000¶Published online: 30 October 2000