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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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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
