Let S denote a compact, connected, orientable surface with genus g and h boundary
components. We refer to S as a surface of genus g
with h holes. Let [Mscr ]S denote
the mapping class group of S, the group of isotopy classes of orientation-preserving
homeomorphisms S → S.
Let G be a group. G is hopfian if every homomorphism from G onto itself is an
automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists
a normal subgroup of finite index in G which does not contain g. Every finitely
generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian
([2, 3]) if every homomorphism ψ G → G
with ψ(G) normal in G and G/ψ(G) cyclic
is an automorphism. As observed in , examples of hopfian groups which are not
hyperhopfian are afforded by the fundamental groups of torus knots.
By a result of Grossman , [Mscr ]S
is residually finite. Since [Mscr ]S is also finitely
generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian.
In this paper, we shall answer a more general question. We say that a group G is
ultrahopfian if every homomorphism ψ: G → G with ψ(G)
normal in G and G/ψ(G)
abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We
shall prove the following result.