Summary: RESIDUAL PROPERTIES OF GRAPH MANIFOLD GROUPS
MATTHIAS ASCHENBRENNER AND STEFAN FRIEDL
To the memory of Bernard Perron (19442008).
Abstract. Let f : M N be a continuous map between closed irreducible
graph manifolds with infinite fundamental group. Perron and Shalen [PS99]
showed that if f induces a homology equivalence on all finite covers, then f is
in fact homotopic to a homeomorphism. Their proof used the statement that
every graph manifold is finitely covered by a 3-manifold whose fundamental
group is residually p for every prime p. We will show that this statement
regarding graph manifold groups is not true in general, but we will show how
to modify the argument of Perron and Shalen to recover their main result.
As a by-product we will determine all semidirect products Z Zd which are
residually p for every prime p.
We say that a group is a p-group if it is finite of order a power of p. (Here and in
the rest of the paper, p will denote a prime number.) We say that a group G is
residually p if for any non-trivial g G there exists a morphism : G P to a
p-group P such that (g) is non-trivial. We say that G is virtually residually p if
there exists a finite index subgroup of G which is residually p. It is not restrictive
to demand that this finite index subgroup is normal in G. (See Section 3 below.)