We prove that the infinite family of asymptotic mapping class groups of surfaces defined by Funar-Kapoudjian and Aramayona-Funar are of type F∞, 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∞. As important examples, we obtain type F∞ 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. © 2024 Real Sociedad Matemática Española.

J. Aramayona was supported by grant PGC2018-101179-B-I00, and acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S and CEX2023-001347-S)”. K.-U. Bux and J. Flechsig were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) 426561549. K.-U. Bux also acknowledges support by the DFG via project 491392403 –TRR 358.

Peer reviewed