Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

TL;DR

This paper proves that certain asymptotic mapping class groups of Cantor manifolds are of type $F_inf$, extending known results to a broader class of manifolds and revealing their homological properties.

Contribution

It establishes the $F_inf$ finiteness property for asymptotic mapping class groups of Cantor manifolds, generalizing previous results and linking to stable homology.

Findings

Asymptotic mapping class groups are of type $F_inf$ under general conditions.

Includes groups containing mapping class groups, automorphism groups, and arithmetic groups.

Homology matches stable homology of classical mapping class groups for certain manifolds.

Abstract

We prove that the infinite family of asymptotic mapping class groups of surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type $F_\infty$, thus answering questions of Funar-Kapoudjian-Sergiescu and Aramayona-Vlamis. As it turns out, this result is a specific instance of a much more general theorem which allows to deduce that asymptotic mapping class groups of Cantor manifolds, also introduced in this paper, are of type $F_\infty$, provide the underlying manifolds satisfy some general hypotheses. As important examples, we will obtain $F_\infty$ asymptotical mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups. In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher--Wahl.