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(ϕi 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 ϕ⊂S 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(ϕsi X,Y)} i=1 ∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.
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Oblatum 4-I-1999 & 19-VII-2000¶Published online: 30 October 2000
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Brock, J. Iteration of mapping classes and limits of hyperbolic 3-manifolds. Invent. math. 143, 523–570 (2001). https://doi.org/10.1007/PL00005799
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DOI: https://doi.org/10.1007/PL00005799