Showing papers on "Free product published in 1998"

Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory

Abstract: Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory Operator-valued stochastic processes and stochastic differential equations Bibliography.

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.

L2-determinant class and approximation of L2-Betti numbers

Abstract: A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L2-Betti numbers. Errata are added, rectifying some unproved statements about "amenable extension": throughout, amenable extensions should be extensions with \emph{normal} subgroups.

Conjugacy separability and free products of groups with cyclic amalgamation

Abstract: A group G is conjugacy separable if whenever x and y are non-conjugate elements of G, there exists some finite quotient of G in which the images of x and y are non- conjugate. It is known that free products of conjugacy separable groups are again conjugacy separable [19, 12]. The property is not preserved in general by the formation of free products with amalgamation; but in [15] a method was introduced for showing that under certain circumstances, the free product of two conjugacy separable groups G1 and G2 amalgamating a cyclic subgroup is again conjugacy separable. The main result of [15] states that this is the case if G1 and G2 are free-by-finite or finitely generated and nilpotent-by-finite. We show here that the same conclusion holds for groups G1 and G2 in a considerably wider class, including, in particular, all polycyclic-by-finite groups. (This answers a question posed by C. Y. Tang, Problem 8.70 of the Kourovka Notebook [7], as well as two questions recently asked by Kim, MacCarron and Tang in G. Kim, J. MacCarron and C. Y. Tang, ‘On generalised free products of conjugacy separable groups’, J. Algebra 180 (1996) 121–135.)

Unification of Independence in Quantum Probability

Abstract: Let , be the conditionally free product of unital free *-algebras , where ϕl, ψl are states on , l∈I. We construct a sequence of noncommutative probability spaces , m∈N, where and , m∈N, , and the states , ϕl are Boolean extensions of ϕl, ψl, l∈I, respectively. We define unital *-homomorphisms such that converges pointwise to *l∈I(ϕl,ψl). Thus, the variables j(m)(w), where w is a word in , converge in law to the conditionally free variables. The sequence of noncommutative probability spaces , where and Φ(m) is the restriction of to , is called a hierarchy of freeness. Since all finite joint correlations for known examples of independence can be obtained from tensor products of appropriate *-algebras, this approach can be viewed as a unification of independence. Finally, we show how to make the m-fold free product into a cocommutative *-bialgebra associated with m-freeness.

Noncommutative joint dilations and free product operator algebras

Abstract: Let An (n = 2, 3, . . . , or n = ∞) be the noncommutative disc algebra, and On (resp. Tn) be the Cuntz (resp. Toeplitz) algebra on n generators. Minimal joint isometric dilations for families of contractive sequences of operators on a Hilbert space are obtained and used to extend the von Neumann inequality and the commutant lifting theorem to our noncommutative setting. We show that the universal algebra generated by k contractive sequences of operators and the identity is the amalgamated free product operator algebra ∗CAni for some positive integers n1, n2, . . . , nk ≥ 1, and characterize the completely bounded representations of ∗CAni . It is also shown that ∗CAni is completely isometrically imbedded in the “biggest” free product C∗-algebra ∗CTni (resp. ∗COni), and that all these algebras are completely isometrically isomorphic to some universal free operator algebras, providing in this way some factorization theorems. We show that the free product disc algebra ∗CAni is not amenable and the set of all its characters is homeomorphic to (C1)1 × · · · × (Ck)1. An extension of the Naimark dilation theorem to free semigroups is considered. This is used to construct a large class of positive definite operator-valued kernels on the unital free semigroup on n generators and to study the class Cρ (ρ > 0) of ρ-contractive sequences of operators. The dilation theorems are also used to extend the operatorial trigonometric moment problem to the free product C∗-algebras ∗CTni and ∗COni .

On the residual nilpotence of free products of free groups with cyclic amalgamation

Abstract: LetG be a free product of free groups with cyclic amalgamation. In this note we prove a criterion for the groupG to be a residually finitep-group. Other residual properties of the groupG are also established.

On the homology of a?free nilpotent group of class?2

Abstract: Let be a free nilpotent group of class 2, and let be a free nilpotent Lie ring of class 2 with the same number of free generators. For a free resolution is constructed which as a graded -module is isomorphic to , where is the group ring of the group and is the exterior algebra of the ring . As a consequence of the basic construction an isomorphism of integral homology is derived.

The cohomology of the Morava stabilizer group ₂ at the prime 3

Abstract: We compute the cohomology of the Morava stabilizer group S2 at the prime 3 by resolving it by a free product Z/3 * Z/3 and analyzing the "4relation module."

The Atlas of Finite Groups - Ten Years on: A survey of symmetric generation of sporadic simple groups

Abstract: Many of the sporadic simple groups possess highly symmetric generating sets which can often be used to construct the groups, and which carry much information about their subgroup structure. We give a survey of results obtained so far. Introduction and motivation This paper is concerned with groups which are generated by highly symmetric subsets of their elements: that is to say by subsets of elements whose set normalizer within the group they generate acts on them by conjugation in a highly symmetric manner. Rather than investigate the behaviour of various known groups, we turn the procedure around and ask what groups can be generated by a set of elements which possesses certain assigned symmetries. It turns out that this approach enables us to define and construct by hand a large number of interesting groups—including many of the sporadic simple groups. Accordingly we let m * n denote C m * C m * – * C m , a free product of n copies of the cyclic group of order m . Let F = T 0 * T 1 * … * T n −1 be such a group, with T i = 〈 t i 〉 ≅ C m . Certainly permutations of the set of symmetric generators T = { t 0 , t 1 , …, t n −1 } induce automorphisms of F . Further automorphisms are given by raising a given t i to a power of itself coprime to m , while fixing the other symmetric generators. Together these generate the group M of monomial automorphisms of F which is a wreath product H r S n , where H r is an abelian group of order r = Φ( m ), the number of positive integers less than m and coprime to it.

Compactly-aligned discrete product systems, and generalizations of 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\to 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 C*_{cov}(P,E) which is universal for covariant representations of E. Our main theorem is a characterization of the faithful representations of C*_{cov}(P,E) when P is the positive cone of a free product of totally-ordered amenable groups.

On rigid monoids with right ACC1

Abstract: We construct a rigid irreflexive monoid with right ACC1 which is not the free product of its group of units and any submonoid. This example disproves a conjecture of Cohn's.

Almost locally free groups and the genus question

Abstract: Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F2)⋂∀∃ in a first-order language Lo appropriate for group theory. It is shown that in every model of Th(F2)⋂∀∃ the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ∃-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F2)⋂∀∃ while the quasi-locally free groups are the ∃-free groups with maximal Abelian subgroups elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound ...

Tensor product construction of 2-freeness

Abstract: From a sequence of m-fold tensor product constructions that give a hierarchy of freeness indexed by natural numbers m we examine in detail the first non-trivial case corresponding to m = 2 which we call 2-freeness. We show that in this case the constructed tensor product of states agrees with the conditionally free product for correlations of order ≤ 4. We also show how to associate with 2-freeness a cocommutative ∗-bialgebra.

Units in Group Rings of Free Products of Prime Cyclic Groups

Abstract: Let G be a free product of cyclic groups of prime order. The structu re of the unit groupU( G) of the rational group ring G is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups ofU( G), up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in G is proved. A strong version of the Tits Alternative for U( G) is obtained as a corollary of the structural result.