Asymptotic mapping class groups of Cantor manifolds and their finiteness properties (with an appendix by Oscar Randal-Williams) (Q6653228)

Asymptotic mapping class groups of surfaces were introduced by \textit{L. Funar} and \textit{C. Kapoudjian} in [Geom. Funct. Anal. 14, No. 5, 965--1012 (2004; Zbl 1078.57021)] with the original motivation of finding natural discrete analogues of the diffeomorphism group of the circle.\N\NIn the paper under review the authors prove that the infinite family of asymptotic mapping class groups of surfaces defined by Funar-Kapoudjian [loc. cit.] and by the first author and \textit{L. Funar} [Mosc. Math. J. 21, No. 1, 1--29 (2021; Zbl 1475.57031)] are of type \(F_{\infty}\), thus answering a problem posed in [\textit{L. Funar} et al., IRMA Lect. Math. Theor. Phys. 17, 595--664 (2012; Zbl 1291.57001)] and a question posed by the first author and \textit{L. Funar} (\textit{loc. cit.}). This result is a specific case of a more general theorem which allows the authors to deduce that asymptotic mapping class groups of certain Cantor manifolds, introduced in this paper, are of type \(F_{\infty}\). As important examples, they 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.\N\NIn addition, for certain types of manifolds, the homology of these asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by \textit{J. L. Harer} [Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)] and by \textit{A. Hatcher} and \textit{N. Wahl} [Geom. Topol. 9, 1295--1336 (2005; Zbl 1087.57003)].