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

Thierry CoulboisMark SapirPascal Weil — 2003

Publicacions Matemàtiques

Let be a pseudovariety of finite groups such that free groups are residually , 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- completions of F(A) and F(B) is also injective. In particular, if is extension-closed, this is the case if and only if φ(F(A)) and its pro- closure in F(B) have the same rank. We examine a number of situations where the injectivity of φ' can be...

Almost all one-relator groups with at least three generators are residually finite

Mark V. SapirIva Špakulová — 2011

Journal of the European Mathematical Society

We prove that with probability tending to 1, a one-relator group with at least three generators and the relator of length n is residually finite, is a virtually residually (finite p -)group for all sufficiently large p , and is coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.

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