We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism. AMS Classification 20E05; 20F28

We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors. 2000 Mathematical Subject Classification: 37A20, 20E05, 82B43, 05C80

Let F be a free group of finite rank. We say that the monomorphism problem in F is decidable if there is an algorithm such that, for any two elements u and v in F , it determines whether there exists a monomorphism of F that sends u to v. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem. Mathematics Subject Classifica...

We prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups. We also prove that it is decidable whether or not a finitely generated submonoid of a free group is graded, and solve the homomorphism and isomorphism problems for graded submonoids of free groups. This generalizes earlier results for submonoids of free monoids. MSC: 20E05; 20...

Let G be any locally compact metrizable group. The main result of this paper, roughly stated, is that if F < G is any finitely generated free group and Γ < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of Γ. If G/Γ is noncompact then we require additional hypotheses that include G = SO(n, 1). MSC: 20E07, 20F65, 20F67, 22D40, 20E05.

Elements of the commutator subgroup of a free group F can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words wg which can be presented by f .g/ distinct Wicks forms, where f .g/ > gŠ. Moreover we may choose these words wg to be square-free. Mathematics Subject Classification (2000). 20E05, 20F10.

abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

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 φ̂ 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...

We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary @T and a free subgroup of G with trivial stabilizer ofw, or there existsw 2 @T and a free subgroup ofG fixingw and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of g...

We prove that the Bianchi groups, that is the groups PSL(2,O) where O is the ring of integers in an imaginary quadratic number field, are good. This is a property introduced by J.P. Serre which relates the cohomology groups of a group to those of its profinite completion. We also develop properties of goodness to be able to show that certain natural central extensions of Fuchsian groups are res...