We study the large-scale geometry of mapping class groups of surfaces of infinite
type, using the framework of Rosendal for coarse geometry of non-locally-compact
groups. We give a complete classification of those surfaces whose mapping class
groups have local
coarse boundedness (the analog of local compactness). When the
end space of the surface is countable or
tame, we also give a classification of those
surfaces where there exists a coarsely bounded generating set (the analog of finite or
compact generation, giving the group a well-defined quasi-isometry type) and
those surfaces with mapping class groups of bounded diameter (the analog of
compactness).
We also show several relationships between the topology of a surface
and the geometry of its mapping class groups. For instance, we show that
nondisplaceable subsurfaces are responsible for nontrivial geometry and can
be used to produce unbounded length functions on mapping class groups
using a version of subsurface projection; while
self-similarity of the space
of ends of a surface is responsible for boundedness of the mapping class
group.
Keywords
big mapping class group, CB generating set, coarsely
bounded sets