Let be a finite alphabet. The Ribes-Zalesskii product theorem states
that,
for any positive integer and any finitely generated subgroups
of the free group on , the set
is closed in the
profinite topology of . Ribes and Zalesskii even proved that ``profinite topology''
may
be replaced with ``pro- topology'' where is any *extension
closed*
pseudovariety of finite groups (under the assumption that the groups
are pro- closed). Recent research by B. Steinberg and myself shows that
this theorem holds under a much weaker assumption on : namely that the
(profinite) Cayley graph of the pro- completion
of
behaves geometrically
*like a tree*. I shall discuss requirements on an inverse system
of -generated finite groups in order that
admits such a ``tree-like'' Cayley graph. It turns out that this happens when
each admits a ``sufficiently good''
co-extension among the groups (this can be made precise). As a
consequence, one
gets that for a pseudovariety of groups *sufficient*
for the free pro- group
to have
a tree-like Cayley graph is that for each group
there exists a cyclic group
for which the wreath product also belongs to ; this is
obviously much weaker than being extension closed.

Sponsored in part by the FCT approved
projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with
the European Community Fund FEDER and by FCT through *Centro de Matemática
da Universidade do Porto.* Also sponsored in part by FCT, the
*Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III*, and by *Caixa Geral de Depósitos*.