Let V be a pseudovariety of finite groups such that free groups are residually V, and let φ: F(A) → F(B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ' of φ between the pro-V completions of F(A) and F(B) is also injective. In particular, if V is extension-closed, this is the case if and only if φ(F(A)) and its pro-V closure in F(B) have the same rank. We examine a number of situations where the injectivity of φ' can be asserted, or at least decided, and we draw a few corollaries.
@article{urn:eudml:doc:41465,
title = {A note on the continuous extensions of injective morphisms between free groups to relatively fre profinite groups.},
journal = {Publicacions Matem\`atiques},
volume = {47},
year = {2003},
pages = {477-487},
zbl = {1064.20028},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41465}
}
Coulbois, Thierry; Sapir, Mark; Weil, Pascal. A note on the continuous extensions of injective morphisms between free groups to relatively fre profinite groups.. Publicacions Matemàtiques, Tome 47 (2003) pp. 477-487. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41465/