Showing papers on "Free product published in 1979"

Groups in which every Finitely Generated Subgroup is almost a Free Factor

Abstract: In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

Chain conditions in free products of lattices with infinitary operations.

Abstract: There are many facts known about the size of subsets of certain kinds in free lattices and free products of lattices. Examples: every chain in a free lattice is at most countable; every "large" subset contains an independent set; if the free product of a set of lattices contains a "long" chain, so does the free product of a finite snbset of this set of lattices. Here we investigate these problems in the setting of a variety V of m-lattices, where m is an infinite regular cardinal. An m-lattice L is a lattice in which for any nonempty set S with lSI

On the SQ-universality of HNN groups

Abstract: A group G is SQ-universal if every countable group can be embedded in a quotient of G. In this paper we give conditions under which HNN groups are SQ-universal. These are then applied to showing the SQ-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL2(C)). Finally some connections are mentioned between SQ-universality and the theory of ends. Introduction. A group G is SQ-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are SQuniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also SQ-universal [10], [11]. Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9]. In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3]. This will be applied in ?3 to showing the SQ-universality of a class of Kleinian groups. Finally we make some observations on the relationship between SQ-universality and the theory of ends. 1. Recall that an HNN group is a group of the form G = i (Li) the free part. Groups of this type were used by Higman, Neumann and Neumann to show the SQ-universality of free groups of rank 2 [4]. The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developed by Karrass and Solitar [5]. Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an SQ-universal quotient is itself SQ-universal we get LEMMA. An HNN group with freq part of rank > 1 is SQ-universal. Received by the editors July 6, 1977. AMS (MOS) subject classifications (1970). Primary 20E30, 20E99; Secondary 20H10.

The Hope property and free products of semigroups

Abstract: The free product of two Hopfian groups (in the category of groups) need not be Hopfian. We prove, by elementary methods, that the free product of two simple Hopfian inverse semigroups is Hopfian. In particular the free product of any two Hopfian groups, in the category of inverse semigroups, is again Hopfian. In fact the same is true in the category of all semigroups.

Free groups, symmetric and reduced products

Abstract: We show that, for any Tychonoff space X with base point θ, the infinite symmetric product SP∞ X of X is a subspace of an abelian group A(X) generated by X. (This clarifies the continuity of the multiplication in SP∞ X.) Furthermore, SP∞ X is a retract of A(X). Analogous results hold for reduced product spaces, with respect to non-abelian groups.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 22 A 99; secondary 54 B 15.

Computation of the homology ofΩ(X∨Y)

Abstract: We compute the homology of Ω(X∨Y) (the loop space of the wedge of the spaces X and Y), in terms of the homogies of ΩX and ΩY. To do this we use the fact that our problem is equivalent to the computation of the homology of the free product of two topological groups in terms of the homologies of the topological groups. We establish a multiple Kunneth formula with coefficients over a Dedekind domain, which is used to prove a Kunneth like formula involves homologies over a Dedekind domain and generalizes similar results with integral or field coefficients. Over a principal ideal domain the formula for a free product is made more specific.

An inductive theory for free products of groups

Abstract: In this paper we apply the ideas introduced in §6 of [1] to free products. While it is easy but tedious to show that the theory of a free product of two groups-not both of order two-becomes undecidable upon adjunction of a binary predicate for equality of syllable length (cf. [2]), we feel that there is hope of finding conditions for the factors that ensure decidability for the theories described below. They are based on a unary operation that picks out the terminal syllable of a word. A natural induction schema plays a crucial role and allows the development of the usual combinatorial machinery. The main purpose of this paper is the publication of some problems whose solutions should shed some new light on Tarski's conjectures.

Discrete free products of two complex cyclic matrix groups

Abstract: Observe that the elements of ftxUf^ have infinite order, while those of ft2Uft3 have finite order. We will prove that whenever A e ft and B e ft do not both have a fixed point on L, then the group (A,B') is the discrete free product (A)*(B'), where B' denotes the transpose of B. The case where both A and B have a fixed point on L is also discussed. We show in addition that if A eft and Beft are real, then (A,B') is the discrete free product (A)*(B') if and only if for every real u,

Coherent group rings and finiteness conditions for CW-complexes

Abstract: We characterize the class of groups G that have the property that if X is any space for which 7TIX G, then X is homotopy equivalent to a space with finite skeleta in the "stable range" if and only if the homotopy groups of X are finitely presented Z[G]-modules in this range. This class of groups includes all finite groups, finitely generated abelian groups, finitely generated nilpotent groups, finitely generated free groups, and free products of any of these. We will say that a group G is coherent if G is finitely presented and if Z [G], the integral group ring of G, is a coherent ring (see below). This class of groups includes, for example, all finite groups, finitely generated abelian groups, finitely generated nilpotent groups, finitely generated free groups, and free products of any of these (see [1] and [4, p. 136]). The following theorem was proved implicitly in [2]. THEOREM 1. Let G be a coherent group. If n is a positive integer and X is a space for which iX 7rvK(G, l)for i 2 for which the conclusions of Theorem 1 are valid, then G is coherent. This of course implies that the conclusions of Theorem 1 are then valid for any positive integer n. I have not been able to characterize the class of groups for which Theorem 1 holds with n = 1; this class certainly includes the coherent groups, but might possibly include some noncoherent groups as well. DEFINITION. A ring R is coherent if every homomorphism between finitely generated free modules has a finitely generated kernel. Received by the editors June 2, 1978. AMS (MOS) subject classifications (1970). Primary 55D99. ? 1979 American Mathematical Society 0002-9939/79/0000-0233/$01.75 368 This content downloaded from 207.46.13.33 on Sun, 20 Nov 2016 04:12:10 UTC All use subject to http://about.jstor.org/terms COHERENT GROUP RINGS AND FINITENESS CONDITIONS 369 The class of coherent rings thus includes the Noetherian rings. Coherent group rings also have important applications in algebraic K-theory (see [5]). Before giving the proof of Theorem 2, we give the following slight refinement of Theorem 1. THEOREM 1'. Let G be a coherent group. If n is a positive integer and X is a space for which giX ri K(G, l) for i F be a homomorphism of Z [G]-modules with F and F free of finite ranks s and t, respectively. We must show that the kernel of f is finitely generated. Let W be the wedge of our K(G, 1) with t copies of the (n + 1)-sphere. Then [6, Theorem 19] n + IW = F as Z [G ]-modules, and we can attach s different (n + 2)-cells to W, with attaching maps determined by the homomorphism f, and obtain a complex X. Now X is a complex with a finite (2n + 1)-skeleton, and TiX-_ giK(G, 1) for i and so H,+2X is a finitely generated Z [G]-module, and the proof of Theorem 2 is complete.

Finite number of double cosets in a free product with amalgamation

Abstract: If H is a finitely generated subgroup of a free group G such that every conjugate of H contains a cyclically reduced word then (G: H) 1. In Theorem 2 by a cyclically reduced word we mean an element w E G of the form w = alb .* * anbn or w = bla, *.* * bnan where ai E A-U, bi E B Uandn > 1. Theorem 1 should be viewed as a generalization of the well-known result of A. Karrass and D. Solitar [2]: If H is a finitely generated subgroup of a free group G and H contains a nontrivial normal subgroup of G then (G: H) < oo. Indeed, if N is such a normal subgroup then xHx D N for all x and N contains a cyclically reduced word. The following result follows immediately from Theorem 2. Received by the editors July 13, 1978. AMS (MOS) subject classifications (1970). Primary 20E05, 20E30; Secondary 05C10, 05C25.