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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.more » « less
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Abstract For every$$n\ge 2$$ , thesurface Houghton group$${\mathcal {B}}_n$$ is defined as the asymptotically rigid mapping class group of a surface with exactlynends, all of them non-planar. The groups$${\mathcal {B}}_n$$ are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some$${\mathcal {B}}_n$$ . As countable mapping class groups of infinite type surfaces, the groups$$\mathcal {B}_n$$ lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that$$\mathcal {B}_n$$ is of type$$\text {F}_{n-1}$$ , but not of type$$\text {FP}_{n}$$ , analogous to the braided Houghton groups.more » « less
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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].more » « less
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Abstract Given a lattice Veech group in the mapping class group of a closed surface , this paper investigates the geometry of , the associated ‐extension group. We prove that is the fundamental group of a bundle with a singular Euclidean‐by‐hyperbolic geometry. Our main result is that collapsing “obvious” product regions of the universal cover produces an action of on a hyperbolic space, retaining most of the geometry of . This action is a key ingredient in the sequel where we show that is hierarchically hyperbolic and quasi‐isometrically rigid.more » « less
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Abstract Given an irreducible, end‐periodic homeomorphism of a surface with finitely many ends, all accumulated by genus, the mapping torus, , is the interior of a compact, irreducible, atoroidal 3‐manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of in terms of the translation length of on the pants graph of . 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.more » « less
