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.

For any finitely generated group G, two complexity functions αG and βG are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of G and the norm of its inverse. Restricting attention to free groups, Fr, the exact asymptotic behaviour of α2 and β2 is computed. For rank r > 3, polynomial lower bounds are provided for αr and βr, and the...

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...

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 g, p ∈ [0↑∞ [, the set of non-negative integers. Let Ag,p denote the group consisting of all those automorphisms of the free group on t[1↑p]∪x[1↑g]∪y[1↑g] which fix the element Π j∈[p↓1] tj Π i∈[1↑g] [xi, yi] and permute the set of conjugacy classes { [tj ] : j ∈ [1↑p]}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others...

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 establish a decomposition of stable homology of automorphism groups of free groups with polynomial contravariant coefficients in term of functor homology. This allows several explicit computations, intersecting results obtained by independant methods by O. Randal-Williams. Classification MSC 2010 : 18A25, 18G15, 20E36, 20J06, 55P42 (18A40, 18E30, 18G40, 55P47). Mots clefs : groupes d’automor...

Let G be a group. An endomorphism φ of G is called rational if there exist a1, . . . , ar ∈ G and h1, . . . , hr ∈ Z, such that φ(x) = (xa1)1 . . . (xar)r for all x ∈ G. We denote by Endr(G) the group of invertible rational endomorphisms of G. In this note, we prove that G is nilpotent of class c (c ≥ 3) if and only if Endr(G) is nilpotent of class c − 1. Mathematics Subject Classification: 20E...