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https://doi.org/10.1007/s00208-023-02751-2

Aramayona, Javier; Bux, Kai-Uwe; Kim, Heejoung; Leininger, Christopher J. (November 2023, Mathematische Annalen)

Abstract For every$$n\ge 2$$

n \geq 2

, thesurface Houghton group$${\mathcal {B}}_n$$

B_{n}

is defined as the asymptotically rigid mapping class group of a surface with exactlynends, all of them non-planar. The groups$${\mathcal {B}}_n$$

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$${\mathcal {B}}_n$$

B_{n}

. As countable mapping class groups of infinite type surfaces, the groups$$\mathcal {B}_n$$

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$$\mathcal {B}_n$$

B_{n}

is of type$$\text {F}_{n-1}$$

F_{n - 1}

, but not of type$$\text {FP}_{n}$$

{FP}_{n}

, analogous to the braided Houghton groups.

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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.

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