Showing papers on "Free product published in 1995"

Free products of compact quantum groups

Abstract: We construct and study compact quantum groups from free products ofC *-algebras. In this connection, we discover two mysterious classes of natural compact quantum groups,A u (m) andA o (m). TheA u (m)'s (respectivelyA o (m)'s) are non-isomorphic to each other for differentm's, and are not obtainable by the ordinary quantization method. We also clarify some basic concepts in the theory of compact quantum groups.

Free Probability Theory: Random Matrices and von Neumann Algebras

Abstract: Independence in usual noncommutative probability theory (or in quantum physics) is based on tensor products. This lecture is about what happens if tensor products are replaced by free products. The theory one obtains is highly noncommutative: freely independent random variables do not commute in general. Also, at the level of groups, this means instead of ℤ n we will consider the noncommutative free group F(n) = ℤ* ⋯ *ℤ or, looking at the Cayler graphs, a lattice is replaced by a homogeneous tree.

Free product von Neumann algebras of type

Abstract: In this paper we will show that most free products of von Neumann algebras with respect to nontracial states produce type IIIA factors (2 #& O) . In addition, for all such 2, examples can be obtained with the component algebras being finite dimensional. Finally, conditions will be given to ensure that these free products will be full factors.

Conjugacy Separability of Generalized Free Products of Certain Conjugacy Separable Groups

Abstract: We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.

Test elements for endomorphisms of free groups and algebras

Abstract: There are two well-known approaches to recognizing automorphisms of a free group, i.e. to distinguishing automorphisms from non-automorphisms. The first one is the “inverse function theorem” of Birman. The second one, the “test element” approach, was originated by Nielsen for the free group of rank 2 and then extended to free groups of arbitrary finite rank by Zieschang, Rosenberger and others. In this note, we establish a direct connection between these two approaches: we associate a special matrix with any element of a free group, and show that an automorphism can be distinguished from a non-automorphism in terms of invertibility of such a matrix associated with a particular single element. Similar results hold for free associative and Lie algebras.

Normal subgroups of free constructions of profinite groups and the congruence kernel in the case of positive characteristic

Abstract: We prove the analogue of the Kurosh subgroup theorem for closed normal subgroups of free constructions of profinite groups and also corresponding analogues of abstract structural results for closed normal subgroups of more general free constructions of profinite groups (amalgamated free products, HNN-extensions). The structure theorem is used to obtain a description of the congruence-kernel C of the arithmetic lattice Γ of the group of K-rational points G=G(K) of a semisimple connected algebraic group G of K-rank 1 over a non-Archimedean local field K.

A type IIIλ factor with core isomorphic to the von Neumann algebra of a free group, tensor B(H)

Abstract: The author constructs a free product of M2(C) with a non-tracial state and L∞(0,1) with Lebesgue measure. He shows that this is a factor of type IIIλ with λ∈(0,1) and that the core of this factor is isomorphic to the tensor product of B(H) and L(F∞), the II1 factor of the free group on infinitely many generators.

Almost free algebras

Abstract: We deal with the class of cardinalsk for which a given variety has an almost free, not free, even not essentially free algebra. The main result is that when enough reflection holds, this behaviour is determined by min{n:n=ω or the variety fail a condition CP n }.

On periodic products of groups

Abstract: Adian introduced periodic n-products of groups which are given by imposing of defining relations of the form An=1 on the free product of groups Gα, α∈I, without involutions. The defining relations An=1 are constructed by a complicated induction which is quite similar to the inductive construction of free Burnside groups due to Novikov and Adian. This periodic n-product of groups Gα, α∈I, has the remarkable property that for every either xn=1 or x is conjugate to an element of Gα for some α. The main result of the article is that this property of periodic n-product can be taken as its definition. This gives a new non-inductive characterization of periodic n-products. An analogous characterization of periodic -products due to Ol’shanskii is also given.

On prosolvable subgroups of profinite free products and some applications

Abstract: In this paper we give a discription of “large” prosolvable subgroups of profinite free products. Based on this result and a result of Herfort-Ribes we show that under quite general hypothesis on the obstruction set relatively projective groups are in fact strongly relatively projective. This in turn is one of the main steps in solving the inverseabsolute Galois problem for such groups.

Projective resolutions for graph products

Abstract: Let Γ be a finite graph together with a group G v at each vertex v . The graph product G (Γ) is obtained from the free product of all G v by factoring out by the normal subgroup generated by for all adjacent v , w . In this note we construct a projective resolution for G (Γ) given projective resolutions for each G v , and obtain some applications.

Commutators in free products

Abstract: Topological methods are used to show that for certain subgroups S of a free product F, if w∈S is a commutator in F, then w is a commutator in S.

Primitive elements in the free product of two finite cyclic groups

Abstract: For the free product 〈s |sa=1〉*〈t|tb=1〉 we give a precise description of the wordsu ins,t which are primitive (i.e. there exists an elementv such thatu,v generate the whole group).

Subgroup separable free products with cyclic amalgamation

Abstract: Using topological methods we give a proof that the free product of two strict subgroup separable groups with infinite cyclic amalgamation is subgroup separable.

Buildings as inner ideal geometries

Abstract: We consider a Chevalley-type group G(V) associated to an integrable representation of a Kac-Moody algebra and show that the associated KacMoody group G(A) is a universal cover for G(V) . This observation strengthens a result of Kac-Peterson on representations of G(A) . It also implies that the building associated to an affine Lie algebra can be realized as an inner ideal geometry. We refer the reader to [4] for details concerning Kac-Moody algebras and their representations. Let S be a finite index set, and let g be the Kac-Moody algebra associated to a generalized Cartan matrix, A = (aij)i, jES* We assume throughout that A is indecomposable. Let h be a Cartan subalgebra for g. Its linear dual is h*. Use rI to designate the set of simple roots, A the set of roots, Are the real roots, and Are the positive real roots of g . Let g = h eaEA ga be its root space decomposition. Fix a set of Chevalley generators for g, {ei, fJ i E S} where ei E gai and fi E g-a, for a1i E II . When 'g is affine, we use (*, *) to designate a standard invariant form. Kac-Moody groups and buildings. We consider two types of groups associated to g and its representations. The first is defined in terms of A without reference to g at all. This is the Kac-Moody group which we designate G(A). It is constructed directly as follows. (This definition is from [8]. See [5] for an axiomatic treatment.) Let G* be the free product of the additive groups ga for a E Are. Denote the canonical injections by la : ga, G*. If (V, d7r) is an integrable representation of g, then we have a representation r*: G* Autc(V) defined by 7C*(la(Xa)) = exp d7r(xa). (Since Xa is locally nilpotent, exp dlC(xa) is a well-defined automorphism of V.) Define G(A) = G*/N* for N* = nker7r* where the intersection is taken over all integrable representations of g. As noted in [5], there is an injective homomorphism (i: SL(2, C) -+ G(A) for each i E S. The set of all Gi = (ai(SL(2, C)) generates G(A) . Induce a topology on G(A) by decreeing that the maps (i be continuous with respect to the usual topology on SL(2, C) . The Kac-Moody group is then Received by the editors March 12, 1993. 1991 Mathematics Subject Classification. Primary 17B67, 20E42. This research was supported in part by NSF Grant No. DMS-9009268. @1995 American Mathematical Society 0002-9939/95 $1.00 + $.25 per page

Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups

Abstract: For a CSA group $G$ and a wide class of abelian groups $A$ we give an explicit construction for the tensor $A$-completion of $G$ using free products with amalgamations. We apply the obtained results to the study of basic properties of $A$-free groups. In particular, canonical and reduced forms of elements in $A$-free groups are introduced, and then commuting and conjugate elements are described.

Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

Abstract: We study the problem of finite approximability with respect to conjugacy of amalgamated free products by a normal subgroup and prove the following assertions. A) IfG is the amalgamated free productG=G1*HG2 of polycyclic groupsG1 andG2 by a normal subgroupH, whereH is an almost free Abelian group of rank 2, thenG is finitely approximate with respect to conjugacy. B) (i) IfG1=G2=L is a polycyclic group andG=G1*HG2 is the amalgamated product of two copies of the groupL by a normal subgroupH, thenG is finitely approximable with respect to conjugacy. (ii) IfG is an amalgamated free productG=G1*HG2 of polycyclic groupsG1 andG2 by a normal subgroupH, whereH is central inG1 orG2, thenG is finitely approximable with respect to conjugacy.

File bases in groups

Abstract: File bases — special systems of group generators generalizing the usual bases of Abelian groups — are considered. Examples of such bases in different group classes are listed. It is shown that if some initial groups have file bases, then direct and free products of these groups as well as their extensions also have such bases. With some additional restrictions imposed, the existence of a file basis in factor groups is proved. A list of problems is offered.

On free products of bicyclic monoids

Abstract: Free products of a collection of bicyclic monoids are investigated, both in the category of monoids and the category of inverse monoids. The structure of these free products is indicated through the use of Green’s relations.

The existence of free products in existence varieties of regular semigroups

Abstract: It is shown that an existence varietyV of regular semigroups contains (all) free products if and only ifV consists solely of locally inverse orE-solid semigroups.

Quasiconvexity and Amalgams

Abstract: We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian groups have property (Q). We also give some applications of our results to one-relator groups and exponential groups.

Free products of matrices

Abstract: In this note we give some new examples of matrix groups which are free products. We use this in our study of the Burau representation in dimension four.