Surface Houghton groups

초록

For every n >= 2$n\ge 2$, the surface Houghton groupBn${\mathcal{B}}_n$ is defined as the asymptotically rigid mapping class group of a surface with exactly n ends, all of them non-planar. The groups Bn${\mathcal{B}}_n$ 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 Bn${\mathcal{B}}_n$. As countable mapping class groups of infinite type surfaces, the groups Bn${\mathcal{B}}_n$ 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 Bn${\mathcal{B}}_n$ is of type Fn-1${\text{F}}_{n-1}$, but not of type FPn${\text{FP}}_n$, analogous to the braided Houghton groups.

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