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Surface mapping class groups are ultrahopfian

Published online by Cambridge University Press:  01 July 2000

MUSTAFA KORKMAZ
Affiliation:
Department of Mathematics, Middle East Technical University, Ankara, 06531, Turkey; e-mail: korkmaz@arf.math.metu.edu.tr
JOHN D. McCARTHY
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A. e-mail: mccarthy@math.msu.edu

Abstract

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 SS.

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 gG 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 GG with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots.

By a result of Grossman [5], [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: GG 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.

Type
Research Article
Copyright
2000 Cambridge Philosophical Society

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