231 research outputs found
Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
We prove that the infinite family of asymptotic mapping class groups of
surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type
, thus answering questions of Funar-Kapoudjian-Sergiescu and
Aramayona-Vlamis.
As it turns out, this result is a specific instance of a much more general
theorem which allows to deduce that asymptotic mapping class groups of Cantor
manifolds, also introduced in this paper, are of type , provide the
underlying manifolds satisfy some general hypotheses.
As important examples, we will obtain asymptotical 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.
In addition, for certain types of manifolds, the homology of our asymptotic
mapping class groups coincides with the stable homology of the relevant mapping
class groups, as studied by Harer and Hatcher--Wahl.Comment: With an appendix by Oscar Randal-Williams. (v3: Rewritten
introduction to include more motivation.) 63 pages, 7 figure
Block mapping class groups and their finiteness properties
A Cantor surface is a non-compact surface obtained by gluing
copies of a fixed compact surface (a block), with boundary
components, in a tree-like fashion. For a fixed subgroup , we
consider the subgroup whose elements
eventually send blocks to blocks and act like an element of ; we refer to
as the block mapping class group with local action
prescribed by . The family of groups so obtained contains the asymptotic
mapping class groups of \cite{SW21a,ABF+21, FK04}. Moreover, there is a natural
surjection onto the family symmetric Thompson groups of Farley--Hughes
\cite{FH15}; in particular, they provide a positive answer to \cite[Question
5.37]{AV20}. We prove that, when the block is a (holed) sphere or a (holed)
torus, is of type if and only if is of type .
As a consequence, for every , has a subgroup of type but
not which contains the mapping class group of every compact
subsurface of .Comment: 21 pages, 1 figur
Effectively Closed Infinite-Genus Surfaces and the String Coupling
The class of effectively closed infinite-genus surfaces, defining the
completion of the domain of string perturbation theory, can be included in the
category , which is characterized by the vanishing capacity of the ideal
boundary. The cardinality of the maximal set of endpoints is shown to be
2^{\mit N}. The product of the coefficient of the genus-g superstring
amplitude in four dimensions by in the limit is an
exponential function of the genus with a base comparable in magnitude to the
unified gauge coupling. The value of the string coupling is consistent with the
characteristics of configurations which provide a dominant contribution to a
finite vacuum amplitude.Comment: TeX, 33 page
Surface Houghton groups
For every , the {\em surface Houghton group} is
defined as the asymptotically rigid mapping class group of a surface with
exactly ends, all of them non-planar. The groups are
analogous to, and in fact contain, the braided Houghton groups. These groups
also arise naturally in topology: every monodromy homeomorphisms of a fibered
component of a depth-1 foliation of closed 3-manifold is conjugate into some
. As countable mapping class groups of infinite type surfaces,
the groups lie somewhere between classical mapping class groups
and big mapping class groups. We initiate the study of surface Houghton groups
proving, among other things, that is of type , but not
of type , analogous to the braided Houghton groups.Comment: 19 pages, 1 figur
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