Luis Ribes

A Kurosh subgroup theorem for free pro--products of pro--groups

Let C be a class of finite groups, closed under finite products, subgroups and homomorphic images. In this paper we define and study free pro-e- products of pro-e- groups indexed by a pointed topological space. Our main result is a structure theorem for open subgroups of such free products along the lines of the Kurosh subgroup theorem for abstract groups. As a consequence we obtain

Pro- p Trees and Applications

One of the main topics of this chapter is the study of ‘combinatorial’ constructions for pro-p groups. We are interested in those groups that can be defined by means of universal properties, such as free pro-p groups, free and amalgamated free products, HNN-extensions, etc. In some sense these are the basic building blocks of pro-p groups. As in the case of abstract groups, such constructions and their properties can be often best understood by studying the action of the group arising from the constructions on a natural ‘tree’ determined by the construction. The Bass—Serre theory of abstract groups acting on trees gives a complete and satisfying description of those groups as fundamental groups of certain graphs of groups. There is a concept of ‘tree’ which is appropriate for the study of pro-p groups, namely the so-called pro-p tree; we define it here and develop its main properties. By contrast with the situation for abstract groups acting on trees, the structure of a pro-p group acting on a pro-p tree is not yet fully clear. But this is one of the reasons that make this area so attractive, for it is still full of open and interesting problems. Yet, one has a considerable number of results about pro-p groups acting on a pro-p trees, and we give an ample sample of them in the first four sections of this chapter.

On automorphisms of free pro-

Let F be a (topologically) finitely generated free pro-p-group, and fi an automorphism of F . If p

p

2 and the order of fi is 2, then there is some basis of F such that fi either fixes or inverts its elements. If p does not divide the order of fi, then the subgroup of F of all elements fixed by fi is (topologically) infinitely generated; however this is not always the case if p divides the order of fi . Let p be a fixed prime number, and let F be a free pro-p-group of finite rank. In this paper we study (continuous) automorphisms of F. The group of automorphisms Aut(F) of F is, in a natural way, a profinite group. In [9J, Lubotzky gives global properties of the group Aut(F). Our interest here is rather more local; we describe properties of certain types of automorphisms. The group Aut(F) contains a pro-p-subgroup of finite index; hence the automorphisms of order prime to p must have finite order. In p

-groups. I

& 2, we show that given an automorphism p of order 2 of a finitely generated pro-p-group G (in particular, a free one), there is a minimal set of generators of G such that p sends each of the generators in that set to itself or its inverse. As a consequence we can describe all the conjugacy classes of involutions of Aut(F): they correspond bijectively to those of GL(n,p), where rankF = n. In [4], Gersten proves that if a is an automorphism of an abstract free group of finite rank, then the elements of the group fixed by a form a subgroup of finite rank also. In contrast, in ?3 we show that for a free pro-p-group F of finite rank, the equivalent result need not hold; in fact we prove that if the order of fi E Aut(F) is not divisible by p, and fi is not the identity, then the subgroup of the elements of F fixed by fi is necessarily infinitely generated (i.e. such a subgroup contains no dense subgroup which is finitely generated as an abstract group). This result depends strongly on the fact that the order of the automorphism does not involve the prime p. In fact, in ?4 we Received by the editors November 17, 1988 and, in revised form, February 13, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20E18; 20F28. i) 1990 American Mathematical Society 0002-9939/90

Profinite completions and isomorphic finite quotients

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Frobenius subgroups of free products of prosolvable groups

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Normalizers in groups and in their profinite completions.

Let G be a group and {G,} be the set of all normal subgroups of finite index in G. Then the set {G/G,, Gob} of finite quotients G/G, of G together with the canonical projections gob : G/G, ---) G/Gb whenever G, c Gb is an inverse system. The inverse limit_ I@ G/G, of this system is called the profinite completion of G and is denoted by G. The group d can also be described as the closure of the image of G under the diagonal mapping d : GA n(G/G,) where G/G, is given the discrete topology and n(G/G,) has the product topology. In this description the elements of 6 are the elements (g,) E n (G/G,) which satisfy @&go) =gb whenever G, c Gb. The object of this paper is to prove the following result:

A WREATH PRODUCT APPROACH TO CLASSICAL SUBGROUP THEOREMS

In this paper we establish the existence of profinite Frobenius subgroups in a free prosolvable productA ∐B of two finite groupsA andB. In this way the classification of solvable subgroups of free profinite groups is completed.

p-Extensions of free pro-p groups

Let R be a finitely generated virtually free group (a finite extension of a free group) and let H be a finitely generated subgroup of R. Denote by R̂ the profinite completion of R and let H̄ be the closure of H in R̂. It is proved that the normalizer NR̂(H̄) of H̄ in R̂ is the closure in R̂ of NR(H). The proof is based on the fact that R is the fundamental group of a graph of finite groups over a finite graph and on the study of the minimal H-invariant subtrees of the universal covering graph of that graph of groups. As a consequence we prove results of the following type: let R be a group that is an extension of a free group by finite solvable group, and let x, y ∈ R; then x and y are conjugate in R if their images are conjugate in every finite quotient of R. Let R be a residually finite abstract group. Then R is embedded naturally in its profinite completion R̂ = lim ←− U∈U R/U, where U denotes the collection of all normal subgroups U of finite index in R. Given a subset X of R, denote its topological closure in R̂ by X̄. This paper is concerned with the following problem: if H is a finitely generated subgroup of R, what is the relationship between the normalizer NR(H) of H in R and the normalizer NR̂(H̄) of H̄ in R̂? Originally this question arose in [10] while studying conjugacy separability in groups that arise as iterations of amalgamated free products of certain groups. In [10] the question is answered when R is a finite extension of a polycyclic group; and then the answer is the desirable one: NR̂(H̄) is the closure in R̂ of NR(H). Perhaps not completely surprising given the nature of polycyclic groups, the proof of that result is ‘arithmetic’ and eventually it relies on number theoretic results and methods. In this paper we deal with the case when R is a finitely generated abstract virtually free group, i.e., a finite extension of a free group Φ. Such a group is residually finite. In fact our results are placed in a more general setting: we Mathematics Subject Classification (2010): 20E18, 20E08.