Showing papers on "Free product published in 1987"

On Closed Subgroups of Free Products of Profinite Groups

Abstract: If G is a free product of a family {Ai}ieI of discrete groups then a subgroup H of G is the free product of a free group F and (Af n H), where a E ~(i), i E I, and ~(i) is a set of representatives of Ai \\ G / H. This is the content of the Kurosh subgroup theorem (KST). Is a similar result true for closed subgroups of free (profinite) products of pro finite groups? (Say, with F projective instead of free.) An answer to this question requires an appropriate definition of a free product over an infinite family of groups. Such a definition has been proposed, by Gildenhuys and Ribes in [3], for groups indexed by compact topological spaces so that the factors are locally equal to each other, except for neighbourhoods of one distinguished point. In spite of the fact that the KST holds for open subgroups of such free products, this definition seems to be too restrictive: if H is a closed subgroup of the free product then the groups Af n H, with a E G, i E I, need not be 'locally equal' to each other (cf. Example 2.4). We propose a very natural generalization of the free product with finitely many factors: an inverse limit of such free products (over an inverse system with mappings that send respective factors again into factors of a free product). This, essentially, also includes the definition of [3]. We do not know whether the analogue of the KST holds for open subgroups of these free products. Nevertheless, if we restrict ourselves to separable groups, we give a satisfactory account of the closed subgroups of the free products. 1. The analogue of the KST does not hold, in general, for closed subgroups of free products (Example 5.5). 2. We define for a pro finite group G the notion of projectivity relative to a given family I of its subgroups (Definition 4.2). We show: 2a. if G is a free product of the groups in I, and H is a closed subgroup of G, then H is projective relative to {fa n HI f E I, a E G}; 2b. conversely, if H is separable and projective relative to ~ then H is a closed subgroup of a free product G of a family I of subgroups such that ~ = {fa n HI rEI, a E G}. 3. Separable relative projective pro-p-groups are in fact free products (Corollary 9.6). Hence we can answer a question of Lubotzky [13, 2.10]: 4. The KST holds for separable closed subgroups of free pro-p-products.

PSL(2, q) AS AN IMAGE OF THE EXTENDED MODULAR GROUP WITH APPLICATIONS TO GROUP ACTIONS ON SURFACES

Abstract: The modular group PSH2, Z), which is isomorphic to a free product of a cyclic groupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed q) tha is at PSUl,n image of the modular group ifq =£ 9. (Here, as usual, q is a prime power.) The extended modular group PGU2, Z)contains PSL{2, Z) with index 2. It has a presentationthe subgroup PSL(2, Z) being generated by UV and VW.A simple group which is an image of PGL(2, Z) is also a Z)n. Fo imagr e of PSU2,many reasons connected with PSU2, q) actions on surfaces (which we discuss in Section4) it is important to know when PSU2, q) is also an image of PGL(2 Z). W, e will prove

On embedding of group rings of residually torsion free nilpotent groups into skew fields

Abstract: It is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field. This is a generalization of J. Lewin's theorem, proven for the case of free groups. Our proof is based on the study of the Malcev-Neumann power series ring K

Free products of lattice-ordered groups

Abstract: We give a highly homogeneous representation for the free product of two nontrivial countable lattice-ordered groups and obtain, as a consequence of the method, that the free product of nontrivial lattice-ordered groups is directly indecomposable and has trivial center.