Let
be an orientable, connected topological surface of infinite type (that is, with infinitely
generated fundamental group). The main theorem states that if the genus
of is finite
and at least
,
then the isomorphism type of the pure mapping class group associated to
, denoted by
, detects the homeomorphism
type of
. As a corollary,
every automorphism of
is induced by a homeomorphism, which extends a theorem of Ivanov from the
finite-type setting. In the process of proving these results, we show that
is residually finite
if and only if
has finite genus, demonstrating that the algebraic structure of
can
distinguish finite- and infinite-genus surfaces. As an independent result, we also show
that
fails to be residually finite for any infinite-type surface
.
In addition, we give a topological generating set for
equipped with the compact-open topology. In particular, if
has at most one end accumulated by genus, then
is
topologically generated by Dehn twists, otherwise it is topologically generated by
Dehn twists along with handle shifts.
Keywords
mapping class groups, infinite-type surfaces, topological
groups
