Summary: Injections of mapping class groups.
CHRISTOPHER J. LEININGER
We construct new monomorphisms between mapping class groups of surfaces. The
first family of examples injects the mapping class group of a closed surface into
that of a different closed surface. The second family of examples are defined on
mapping class groups of once-punctured surfaces and have quite curious behaviour.
For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither
of these two types of phenomena were previously known to be possible although
the constructions are elementary.
57M07; 32G15, 57R50
Let g,k be the closed orientable surface of genus g with k marked points and
Homeo(g,k) the group of homeomorphisms of g,k which map the set of marked
points to itself. If we endow Homeo(g,k) with the compact open topology then
Homeo0(g,k), the connected component of the identity, is the normal subgroup con-
sisting of those elements which are isotopic to the identity relative to the marked points.
The (extended) mapping class group of g,k is the quotient group
Mod(g,k) = Homeo(g,k)/Homeo0(g,k).