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

• Javier Aramayona ^[1] ; Kai-Uwe Bux ^[2] ; Jonas Flechsig ^[2] ; Nansen Petrosyan ^[3] ; Xiaolei Wu ^[4]
  1. [1] Instituto de Ciencias Matemáticas

     Instituto de Ciencias Matemáticas

     Madrid, España

     
  2. [2] Bielefeld University

     Bielefeld University

     Kreisfreie Stadt Bielefeld, Alemania

     
  3. [3] University of Southampton

     University of Southampton

     GB.ENG.M4.24UJ, Reino Unido

     
  4. [4] Fudan University

     Fudan University

     China

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 6, 2024, págs. 2003-2072
• Idioma: inglés
• DOI: 10.4171/RMI/1502
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• Resumen

  • 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.