Asymptotic mapping class groups of Cantor manifolds and their finiteness properties release_v3e22q5tb5bwphkop6ffdfblve

by Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig, Nansen Petrosyan, Xiaolei Wu

Released as a article .

2021  

Abstract

A Cantor manifold π’ž 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, Dehornoy, and Funar- Kapoudjian, we introduce the asymptotic mapping class group ℬ of π’ž, whose elements are proper isotopy classes of self-diffeomorphisms of π’ž 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 π’ž. 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 F_∞ and that 𝔛 is CAT(0); more concretely, our methods apply for example when Y is diffeomorphic to π•Š^1 Γ—π•Š^1, π•Š^2Γ—π•Š^1, or π•Š^n Γ—π•Š^n for nβ‰₯ 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 β‰…π•Š^2 or π•Š^1 Γ—π•Š^1, our result gives a positive answer to <cit.> and <cit.>. In addition, for Yβ‰…π•Š^1 Γ—π•Š^1 or π•Š^2Γ—π•Š^1, the homology of ℬ coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher–Wahl.
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Date   2021-10-11
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arXiv  2110.05318v1
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