 
Summary: Grothendieck's problem for 3manifold groups
D. D. Long
& A. W. Reid
May 12, 2010
For Fritz Grunewald
1 Introduction
The profinite completion of a group is the inverse limit of the directed system of finite quotients
of , and we shall denote this profinite group by ^. As is wellknown, if is residually finite then
injects into ^. In [19] the following problem was posed by Grothendieck (where it is pointed out
that it is natural to assume the groups are residually finite):
Let u : H G be a homomorphism of finitely presented residually finite groups for which the
extension ^u : ^H ^G is an isomorphism. Is u an isomorphism?
This problem was solved in the negative by Bridson and Grunewald in [6] who produced many
examples of groups G, and proper subgroups u : H G for which ^u is an isomorphism, but u is
not. The method of proof of [6] was a far reaching generalization of an example of Platonov and
Tavgen [35] that produced finitely generated examples that answered Grothendieck's problem in the
negative (see also [5]).
Notice that if ^H ^G is an isomorphism, then the composite homomorphism H ^H ^G
is an injection. Hence, H G must be an injection. Therefore, Grothendieck's Problem reduces
to the case where H is a subgroup of G, and the homomorphism is inclusion. Henceforth, we will
