Asymptotic mapping class groups of Cantor manifolds and their finiteness properties (with an appendix by Oscar Randal-Williams)

Abstract

We prove that the infinite family of asymptotic mapping class groups of surfaces defined by Funar–Kapoudjian and Aramayona–Funar are of type $F_{\infty }$, thus answering a problem of Funar–Kapoudjian–Sergiescu and a question of Aramayona–Funar. This result is a specific case of a more general theorem which allows us to deduce that asymptotic mapping class groups of certain Cantor manifolds, also introduced in this paper, are of type $F_{\infty }$. As important examples, we obtain type $F_{\infty }$ asymptotic 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.