Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
Javier Aramayona
ICMAT (España)
Abstract:
(Joint work with K.-U. Bux, J. Flechsig, N. Petrosyan, X. Wu) 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 braided Thompson groups, we introduce the asymptotic mapping class group of 
\[C\]
, whose elements are proper isotopy classes of self-diffeomorphisms of 
\[C\]
that are ”eventually trivial.” This group 
\[B\]
happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of 
\[C\]
.
\[B\]
 acts on a contractible cube complex 
\[X\]
of infinite dimension, and we may try to use the action to determine finiteness properties of 
\[B\]
. In well-behaved cases, 
\[X\]
is 
\[CAT(0)\]
and 
\[B\]
is of type 
\[F^{\infty}\]
; more concretely, the methods apply when 
\[Y\]
is a 2-dimensional torus, or 
\[S^2\times S^1\]
, or 
\[S^n\times S^n\]
for 
\[n\]
n at least 3. In these cases, the group 
\[B\]
contains, respectively, the mapping class groups of every compact surface with boundary, the automorphism groups of finitely generated free groups, or an infinite familiy of arithmetic symplectic or orthogonal groups.
In particular, the cases where 
\[Y\]
is a sphere or a torus in dimension 2 yields a positive answer to a question of Funar-Kapoudjian-Sergiescu. In addition, we find cases where the homology of 
\[B\]
coincides with the stable homology of the relevant mapping class groups.