A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups

Autores: Thierry Coulbois, Mark V. Sapir, Pascal Weil
Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 47, Nº 2, 2003, págs. 477-488
Idioma: inglés
DOI: 10.5565/publmat_47203_10
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Resumen
- 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 ˆϕ 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 ˆϕ can be asserted, or at least decided, and we draw a few corollaries.