Browsing by Author "Leininger, Christopher"

Now showing 1 - 4 of 4
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    Degeneration of flat metrics from k–differentials
    (2025-04-23) Zhang, Zhiyi; Leininger, Christopher
    This thesis explores the space of flat metrics sitting in the space of geodesic cur- rents. The space of geodesic currents, considered as the completion of weighted closed curves, was first introduced by Bonahon to study hyperbolic 3–manifolds, and then used by him to provide an alternative description of Thurston’s compactifica- tion of Teichmu ̈ller space. This space has since played a key role in the study of geometric structures on surfaces and their degenerations. We focus on the space of flat metrics from k–differentials on finite–type surfaces and its embedding into the space of geodesic currents. With this embedding, we describe a compactification of the space of flat metrics coming from k–differentials, for all k, where the boundary points are the mixed structures—currents that are a flat metric on a subsurface and a measured lamination on the complement. This generalizes the results of Duchin– Leininger–Rafi for the case when k = 2 and Ouyang–Tamburelli for the case when k = 3, 4.
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    End-periodic homeomorphisms and volumes of mapping tori
    (Wiley, 2023) Field, Elizabeth; Kim, Heejoung; Leininger, Christopher; Loving, Marissa; Mathematics
    Given an irreducible, end-periodic homeomorphism f:S→S$f: S \rightarrow S$ of a surface with finitely many ends, all accumulated by genus, the mapping torus, Mf$M_f$, is the interior of a compact, irreducible, atoroidal 3-manifold M¯f$øverlineM_f$ with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of M¯f$øverlineM_f$ in terms of the translation length of f$f$ on the pants graph of S$S$. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.
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    Pseudo-Anosov subgroups of general fibered 3–manifold groups
    (American Mathematical Society, 2023) Leininger, Christopher; Russell, Jacob; Mathematics
    We show that finitely generated and purely pseudo-Anosov subgroups of fundamental groups of fibered 3–manifolds with reducible monodromy are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. Combined with results of Dowdall–Kent–Leininger and Kent–Leininger–Schleimer, this establishes the result for the image of all such fibered 3–manifold groups in the mapping class group.
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    A universal Cannon-Thurston map and the surviving curve complex
    (American Mathematical Society, 2022) Gültepe, Funda; Leininger, Christopher; Pho-on, Witsarut; Mathematics
    Using the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon-Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer [Comment. Math. Helv. 86 (2011), pp. 769–816].