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Schottky Subgroups of Mapping Class Groups and the Geometry of Surface-by-Free Groups

In: Rigidity in Dynamics and Geometry

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  • Lee Mosher

    (Rutgers University, Department of Mathematics)

Abstract

This is an expanded version of a conference lecture on groups ΓΣF of the form $$1 \to {\pi _1}\sum \to {\Gamma _{\sum , }}F \to F \to 1$$ where Σ is a closed, oriented surface of genus ≥ 2 and F is a free subgroup of rank ≥ 2 in the mapping class group MCG(Σ) = Out(π1Σ). In joint work with Benson Farb [FM01], [FM00a]we characterize when ΓΣF is word hyperbolic, and when it is, we prove that ΓΣF is quasi-isometrically rigid in a very strong sense. These results require a study of stable quasi-geodesics in Teichmüller space [Mos01]. This study also applies to give a proof of Minsky’s Theorem, according to which Thurston’s ending lamination conjecture holds in the case of injectivity radius bounded away from zero.

Suggested Citation

  • Lee Mosher, 2002. "Schottky Subgroups of Mapping Class Groups and the Geometry of Surface-by-Free Groups," Springer Books, in: Marc Burger & Alessandra Iozzi (ed.), Rigidity in Dynamics and Geometry, pages 309-319, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04743-9_16
    DOI: 10.1007/978-3-662-04743-9_16
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