Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
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by
Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig, Nansen Petrosyan, Xiaolei Wu
2021
Abstract
A Cantor manifold π 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 β¬
of π, whose elements are proper isotopy classes of
self-diffeomorphisms of π that are eventually trivial. The
group β¬ happens to be an extension of a Higman-Thompson group by a
direct limit of mapping class groups of compact submanifolds of π.
We construct an infinite-dimensional contractible cube complex π
on which β¬ acts. For certain well-studied families of manifolds, we
prove that β¬ is of type F_β and that π is CAT(0); more concretely, our methods apply for example when Y is
diffeomorphic to π^1 Γπ^1, π^2Γπ^1, or π^n Γπ^n for nβ₯ 3. In these cases,
β¬ 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^2 or π^1 Γπ^1,
our result gives a positive answer to <cit.> and
<cit.>. In addition, for Yβ
π^1 Γπ^1 or π^2Γπ^1, the homology of β¬
coincides with the stable homology of the relevant mapping class groups,
as studied by Harer and HatcherβWahl.
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