Surface Houghton groups

Abstract

AbstractFor every $$n\ge 2$$ n ≥ 2 , the surface Houghton group$${\mathcal {B}}_n$$ 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 $${\mathcal {B}}_n$$ 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.