Showing papers on "Free product published in 1999"

Representations of compact quantum groups and subfactors

Abstract: We associate Popa systems (= standard invariants of subfactors) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be ``represented'' on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. We prove a Kesten type result for the co-amenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.

Mapping tori of free group automorphisms are coherent

Abstract: The mapping torus of an endomorphism ' of a group G is the HNNextensionG⁄G with bonding maps the identity and '. We show that a mapping torus of an injective free group endomorphism has the property that its flnitely generated subgroups are flnitely presented and, moreover, these subgroups are of flnite type.

Amalgamated free product over Cartan subalgebra

Abstract: Amalgamated free products of von Neumann algebras were first used by S. Popa ([26]) to construct an irreducible inclusion of (non-AFD) type II1 factors with an arbitrary (admissible) Jones index. Further investigation in this direction was made by K. Dykema ([10]) and F. Rădulescu ([27, 29]) based on Voiculescu’s powerful machine ([40, 41, 44]), and F. Boca ([4]) discussed the Haagerup approximation property, where only finite von Neumann algebras were dealt with. On the other hand, type III factors arising as free products (over C) were studied by L. Barnett ([3]), K. Dykema ([9, 11]), F. Rădulescu ([28]), and very recently by D. Shlyakhtenko ([33]). However, amalgamated free products in the type III setting have never been seriously investigated so far. The main purpose of the paper is to take a first step towards investigation on amalgamated free products in the type III setting. A construction of amalgamated free products of arbitrary von Neumann algebras has never been (at least explicitly) given in the literature (see [29, 44] in the type II1 case), and hence we present such a construction in §2. Our construction requires (faithful) normal conditional expectations onto a common subalgebra, and the concept of bimodules is useful. We mainly study the amalgamated free product of non-type I factors A,B over their common Cartan subalgebra D:

Simplicity and the stable rank of some free product C*-algebras

Abstract: A necessary and sufficient condition for the simplicity of the C*-algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C*-algebras are proved.

Embeddings of reduced free products of operator algebras

Abstract: Given reduced amalgamated free products of C$^*$-algebras, $(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_i\to D_i$. This result is extended to show the existance of the reduced amalgamated free product of certain classes of unital completely positive maps. Analogues of the above mentioned results are proved for von Neumann algebras.

Automatic monoids and change of generators

Abstract: The main result of this paper establishes invariance under change of generators for automatic structures for monoids (a property that is well known to hold for automatic groups but fails for semigroups). This result is then applied to show that if a free product of two monoids is automatic, then so are both the free factors. Finally, the difference between automatic structures, in terms of monoid generating sets and semigroup generating sets, is discussed.

On Dehn functions of free products of groups

Abstract: In this article we show that the Dehn function of a nontrivial free product of groups is equivalent to its subnegative closure. Let P = 〈Σ | R 〉 be a finite presentation of a group G. Suppose that a word w over Σ is equal to 1 in G. By van Kampen’s Lemma [6], there exists a diagram over P such that w is a boundary label of this diagram. By k(w) we denote the smallest number of cells in such a diagram. In other words, k(w) is the smallest number with the following property: w is equal in the free group over Σ to a product of k(w) conjugates of the defining relators from R or their inverses. A function f : N → N is called the Dehn function of the presentation P whenever f(n) = max k(w), where maximum is taken over all words such that w equals 1 in G and |w| ≤ n. Let f , g be functions from N to N. By definition, f g means that there exists a positive integer C such that f(n) ≤ Cg(Cn)+Cn for all n. This relation induces an equivalence relation on the set of functions from N into itself: f ' g if and only if f g and g f . It is well known (see [7], [1], [4]) that if G is a finitely presented group and f , g are the Dehn functions of two finite presentations of G, then f ' g. In this paper, we shall not distinguish between equivalent functions. Thus we will speak about the Dehn function of a finitely presented group G. Following Brick [3], we say that a function f : N → N is subnegative whenever f(m) + f(n) ≤ f(m + n) for all m, n ∈ N. For every function f : N → N one can define a function f̄(n) by the following formula: f̄(n) = max (f(n1) + f(n2) + · · · f(nr)), (1) where maximum is taken for all r ≥ 1, and n1, . . . , nr ∈ N such that n1+ · · ·+nr = n. It is easy to see that f̄(n) is the smallest subnegative function which is greater than or equal to f(n). The function f̄ is said to be the subnegative closure of f . The subnegative property plays an important role in [2], where it is proved that the class of Dehn functions is very large. Roughly speaking, every relatively fast computable subnegative function ≥ n is the Dehn function of a finitely presented group. More precisely, if T (n) is a subnegative time function of a nondeterministic Received by the editors June 27, 1997. 1991 Mathematics Subject Classification. Primary 20F32; Secondary 57M07. The research of the first author was supported in part by grants from the ISF and the Russian Foundation for Fundamental Researches, grant no. 96–01–00420. The research of the second author was supported in part by NSF grants. c ©1999 American Mathematical Society

On the intersection of finitely generated subgroups in free products of groups

Abstract: A subgroup H of a free product of groups Gα, α∈ I, is called factor free if for every and β ∈ I one has S H S-1∩ Gβ = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote , where r(K) is the rank of K. It is proven that if H, K are finitely generated factor free subgroups of a free product then . It is also shown that the inequality of Hanna Neumann conjecture on subgroups of free groups does not hold for factor free subgroups of free products.

On the structure of two-generated hyperbolic groups

Abstract: We show that every two-generated torsion-free one-ended word-hyperbolic group has virtually cyclic outer automorphism group. This is done by computing the JSJ-decomposition for two-generator hyperbolic groups. We further prove that two-generated torsion-free word-hyperbolic groups are strongly accessible. This means that they can be constructed from groups with no nontrivial cyclic splittings by applying finitely many free products with amalgamation and HNN-extensions over cyclic subgroups.

On 2-Generator Subgroups of SO(3)

Abstract: We classify all subgroups of SO(3) that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of 7r. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by 7r and 7r/2. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.

Shmel’kin embeddings for abstract and profinite groups

Abstract: The Magnus embedding is well known: given a group A=F/R, where F is a free group, the group F/[R, R] can be represented as a subgroup of a semidirect product AT, where T is an additive group of a free Z A-module. Shmel’kin genralized this construction and found an embedding for F/V(R), where V(R) is the verbal subgroup of R corresponding to a variety V. Later, he treated F as a free product of arbitrary groups, and on condition that R is contained in a Cartesian subgroup of the product, pointed out an embedding for F/V(R). Here, we combine both these Shmel’kin embeddings and weaken the condition on R, by assuming that F is a free product of groups Ai (ieI) and a free group X, and that its normal subgroup R has trivial intersection with each factor Ai. Subject to these conditions, an embedding for F/V(R) is found; we cell it the generalized Shmel’kin embedding. For the case where V is an Abelian variety of groups, a criterion is specified determining whether elements of AT belong to an embedded group F/V(R). Similar results are proved also for profinite groups.

Cores of hyperbolic 3-manifolds and limits of Kleinian groups II

Abstract: Troels Jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in 'most' cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups.

A Functional Cryptosystem Using a Group Action

Abstract: The main purpose of this paper is to examine applications of group theoretical concepts to cryptography. We construct a backward deterministic system employing the action of the modular group on the upper half plane and the amalgamated free product structure of the group. We invent a geometrical algorithm that finds the normal form of an element of the modular group effectively. This algorithm makes our backward deterministic system tractable. Using the backward deterministic system, we invent a public-key cryptosystem in terms of a functional cryptosystem.

COMPACTLY-ALIGNED DISCRETE PRODUCT SYSTEMS, AND GENERALIZATIONS OF ${\mathcal O}_\infty$

Abstract: The universal C*-algebras of discrete product systems generalize the Toeplitz–Cuntz algebras and the Toeplitz algebras of discrete semigroups. We consider a semigroup P which is quasi-lattice ordered in the sense of Nica, and, for a product system p : E→ P, we study those representations of E, called covariant, which respect the lattice structure of P. We identify a class of product systems, which we call compactly aligned, for which there is a purely C*-algebraic characterization of covariance, and study the algebra which is universal for covariant representations of E. Our main theorem is a characterization of the faithful representations of when P is the positive cone of a free product of totally-ordered amenable groups.

Exactness of reduced amalgamated free product C*-algebras

Abstract: Some completely positive maps on reduced amalgamated free products of C*-algebras are constructed; these allow a proof that the class of exact unital C*-algebras is closed under taking reduced amalgamated free products. Consequently, the class of exact discrete groups is closed under taking amalgamated free products.

Positive definite kernels on free product semigroups and universal algebras

Abstract: A method to construct positive deenite operator-valued kernels on free product semigroups amalgamated over the identity is presented. A complete description of the structure of positive deenite Toeplitz kernels on free products of semigroups is given.

Minimal growth of non-Hopfian free products

Abstract: We provide examples of finitely generated groups of uniform exponential growth whose minimal growth is not realized by any generating set: namely, all non-Hopfian free products of groups have this property. This result stems from growth tightness of free products: that is, the exponential growth rate of every nontrivial free product, different from Z2 * Z2, its strictly greater than the growth rate of any of its proper quotients.

Free probability on algebras with infinitely many states

Abstract: We study noncommutative probability spaces endowed with infinite sequences of states. Following ideas of Cabanal-Duvillard we extend the notion of conditional freeness. Free product of such spaces is justified by constructing an appropriate ⋆-representation. Finally, we provide limit theorems and describe the sequences of orthogonal polynomials related to the limit measures.

The orthogonal polynomials for a linear sum of a free family of projections

Abstract: We calculate the probability measures for some linear combinations of a free family of projections by using Voiculescu's R-transform for free additive convolution in the free probability theory The measures we give in this paper include many examples that have been calculated, for instance, the spectral measure for the adjacency operators of the infinite distance regular graphs, and the Plancherel measures associated with free product groups Furthermore, we find the recursive relations for the orthogonal polynomials associated with them