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

Abstract

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 extensionφ 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 ofφ can be asserted, or at least decided, and we draw a few corollaries.In this paper, we are interested in the pro-V topologies on finitely generated free groups, where V is a pseudovariety of groups (a class of finite groups closed under taking subgroups, quotients and finite direct products). These topologies were introduced in the 1950s by Hall. When V is the class of all finite groups, the finite index subgroups are exactly the open subgroups, and Hall proved [6] that every finitely generated subgroup is closed. More recent papers (Ribes and Zalesskiȋ [9], Margolis, Sapir and Weil [7], Weil [12]) focused on the problem of effectively computing the pro-V closure Cl V (H) of a given finitely generated subgroup H of a free group. It is known for instance that if V is extension-closed, then Cl V (H) has finite rank, at most equal to the rank of H [9]. In general, a finite rank subgroup may have an infinite rank closure (e.g. if V is the pseudovariety of finite abelian groups), or it may be the case that if H is a finite rank subgroup, then its closure 2000 Mathematics Subject Classification. 20E18, 20E05. Key words. Profinite topology, rank of subgroups of free groups, monomorphisms between free groups. This paper was prepared while the second author was an invited Professor at Université Bordeaux-1. The third author also acknowledges partial support from 1999 INTAS grant number 1224 Combinatorial and geometric theory of groups and semigroups and its applications to computer science.

Freeness of Schützenberger groups of primitive substitutions

“…We provide a proof for the sake of completeness. We chose to rely on [22,25,26], but let us mention that results from [13] could be used as well. Alternatively, one could adapt the proof of [7, Proposition 4.6.5], which can be partly traced back to [1, Lemma 4.2].…”

Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms

“…Proof. Let us fix i from 1 to n. The polynomials x i , f i , f (2) i , ..., f (n) i are algebraically dependent over F q . This means that…”

“…Injective endomorphisms of free groups that have injective extensions in p-adic (resp. pro-solvable, and many other profinite) topologies of a free group are completely described in [2].…”