Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Abstract

A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing classical families of groups due to Brin [10], Dehornoy [18] and Funar–Kapoudjian [21, 23, 1], we introduce the asymptotic mapping class group B of C, whose elements are proper isotopy classes of selfdiffeomorphisms of C that are eventually trivial. The group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C. We construct an infinite-dimensional contractible cube complex X on which B acts. For certain well-studied families of manifolds, we prove that B is of type F8 and that X is CATp0q; more concretely, our methods apply for example when Y is diffeomorphic to S 1 ˆS 1 , S 2 ˆS 1 , or S n ˆS n for n ě 3. In these cases, B contains, respectively, the mapping class group of every compact surface with boundary; the automorphism group of the free group on k generators for all k; and an infinite family of (arithmetic) symplectic or orthogonal groups. In particular, if Y – S 2 or S 1 ˆ S 1 , our result gives a positive answer to [24, Problem 3] and [3, Question 5.32]. In addition, for Y – S 1 ˆ S 1 or S 2 ˆ S 1 , the homology of B coincides with the stable homology of the relevant mapping class groups, as studied by Harer [31] and Hatcher– Wahl [39].

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ORCID for Nansen Petrosyan: orcid.org/0000-0002-2768-5279

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