Showing papers on "Free product published in 1992"
Book•
01 Jan 1992
TL;DR: Free products Free random variables in noncommutative probability theory Free harmonic analysis Random matrices and asymptotic freeness Free product factors Free product factor as mentioned in this paper Free products
Abstract: Free products Free random variables in noncommutative probability theory Free harmonic analysis Random matrices and asymptotic freeness Free product factors.
322 citations
TL;DR: In this article, a characterization of C *-algebras that have a separating family of finite-dimensional representations is given, which makes possible a solution to a problem posed by Goodearl and Menaul.
Abstract: Our main theorem is a characterization of C*-algebras that have a separating family of finite-dimensional representations. This characterization makes possible a solution to a problem posed by Goodearl and Menaul. Specifically, we prove that the free product of such C*-algebras again has this property.
101 citations
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TL;DR: In this article, the free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor.
Abstract: The free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$. The finite dimensional part depends on the minimal projections of the original algebras and the "dimension", r, of the free group factor part is found using the notion of free dimension. For discrete amenable groups $G$ and $H$ this implies that the group von Neumann algebra $L(G*H)$ is an interpolated free group factor and depends only on the orders of $G$ and $H$.
101 citations
06 Apr 1992
TL;DR: It is established here that it is decidable whether a rational set of a free partially commutative monoid is recognizable or not if and only if the commutation relation is transitive.
Abstract: It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized finite automata.
71 citations
TL;DR: A short proof is given of a result of J.W. Morgan and I. Morrison that describes the fundamental group of the Hawaiian earring, which is a countably infinite union of circles that are all tangent to a single line at the same point.
Abstract: The Hawaiian earring is a topological space which is a countably infinite union of circles, that are all tangent to a single line at the same point, and whose radii tend to zero. In this note a short proof is given of a result of J.W. Morgan and I. Morrison that describes the fundamental group of this space. It is also shown that this fundamental group is not a free group, unlike the fundamental group of a wedge of an arbitrary number of circles.
61 citations
TL;DR: In this paper, the authors show that the 2-dimensional Cremona group Cr 2 =Aut k (X, Y) acts on a simplicial complex C, which has as vertices certain models in the function field k(X,Y).
Abstract: We show that the 2-dimensional Cremona group Cr 2 =Aut k (X, Y) acts on a 2-dimensional simplicial complex C, which has as vertices certain models in the function field k(X, Y). The fundamental domain consists of one face F. This yields a structural description of Cr 2 as an amalgamation of three subgroups along pairwise intersections. The subgroup GA 2 =Aut k k[X, Y] (integral Cremona group] acts on C by restriction. The face F has an edge E such that the GA 2 translates of E form a tree T. The action of GA 2 on T yields the well-known structure theory for GA 2 as an amalgamated free product, using Serre's theory of groups acting on trees
29 citations
21 citations
TL;DR: In this paper, the authors define a highly transitive permutation group (X,G) which is n-transitive for all n ≥ 1, provided that the point-stabilizer Stab (S) acts transitively on X-S for some n > 1.
Abstract: A concrete classification was recently given by Peter Neumann and Samson Adeleke [1] of all the primitive Jordan (permutation) groups (X, G) which are not highly transitive, that is, such that G acts w-transitively, but not (n+ Intransitively on X for some n > 1. Such a permutation group (X, G) is called a Jordan group provided that there exists S c X with \\S\\ ^ n +1 and \\X-S\\ ^ 2 such that the point-stabilizer Stab (S) = {ge G \\ (s)g = s for all se S} acts transitively on X— S. Here we shall define a highly transitive permutation group (X,G) (that is, (X,G) is n-transitive for all n ^ 1) to be a Jordan group provided there exists S ^ X such that both S and X—S are infinite and Stab (S) acts highly transitively on X— S (and we shall call X— S a highly transitive Jordan set of (X, G)). Here we shall prove the following.
21 citations
TL;DR: In this article, the growth series of the groups {x,y\y~ 1 x p y=x p ) and (x, y\x p = />, p > 2, with respect to generators {x and y} was computed using minimal normal forms obtained by informal use of judiciously chosen rewrite rules.
Abstract: Here we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups {x,y\y~ 1 x p y=x p ) and (x, y\x p = /> , p > 2, with respect to generators {x, y} . This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group
15 citations
TL;DR: In this article, the Pukanszky invariant of the abelian algebra A =(A ∨ JAJ) is computed for the free product of N groups having order k ⩽ N and A is the maximal subalgebra of the group von Neumann algebra L (G), called the radial algebra of G.
14 citations
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TL;DR: In this paper, the interpolated free group factors L(F_r), 1 < r <= \infty, are defined and proofs of their properties with respect to compression by projections and taking free products are proved.
Abstract: The interpolated free group factors L(F_r), 1 < r <= \infty, are defined and proofs of their properties with respect to compression by projections and taking free products are proved. Hence it follows that all the free group factor are isomorphic to each other or none of them are. These factors were defined and these properties were proved independently by F. Radulescu, and those given in this paper are equivalent, but use different techniques. Specifically, we develop algebraic techniques that allow us to show that R*R = L(F_2), where R is the hyperfinite II_1 factor.
TL;DR: In this paper, it was shown that if C3 occurs as a factor in the free product decomposition of G and if C2 is either not present or occurs to an even power, then Mn 0 0 mod 3 if and only if n -2 mod 4.
Abstract: Let G be the free product of finitely many cyclic groups of prime order. Let Mn denote the number of subgroups of G of index n. Let Cp denote the cyclic group of order p, and Cpk the free product of k cyclic groups of order p . We show that Mn is odd if C4 occurs as a factor in the free product decomposition of G. We also show that if C3 occurs as a factor in the free product decomposition of G and if C2 is either not present or occurs to an even power, then Mn 0 0 mod 3 if and only if n -2 mod 4. If, on the other hand, C3 occurs as a factor, and C2 also occurs as a factor, but to an odd power, then all the Mn are -1 mod 3. Several conjectures are stated.
TL;DR: It is proved that if G is a finitely presented group acting freely on an R tree and Λ is a corresponding set of pseudogroup generators, the authors're in one of the following situations: either G splits as a free product with a noncyclic free abelian summand, or Λ can be reduced to an interval exchange by normalizing and removing a finite number of dead ends.
Abstract: Let be a pseudogroup defined on a tree Z, and let Λ be a finite set of generators for . The reduced fundamental group (Λ) of Λ is defined here. I give a new and experimentally inspired proof of a result of Levitt : If (Λ) is a free group, there exists a finite set of generators Ψ for such that is free on the set Ψ. If Ψ has no dead ends, it is an interval exchange. Like Gaboriau, Levitt and Paulin [Gaboriau et al, 19921. I prove that if G is a finitely presented group acting freely on an R tree and Λ is a corresponding set of pseudogroup generators, we're in one of the following situations: either G splits as a free product with a noncyclic free abelian summand, or Λ can be reduced to an interval exchange by normalizing and removing a finite number of dead ends, or the process of removing dead ends from Λ does not terminate in a finite number of steps.
TL;DR: In this paper, the residual finiteness of free products with amalgamations and AW-extensions of finitely generated nilpotent groups was studied in terms of certain conditions satisfied by the associated subgroups.
Abstract: We study the residual finiteness of free products with amalgamations and //AW-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNNextensions). 1991 Mathematics subject classification (Amer. Math. Soc): 20 E 06, 20 E 26.
TL;DR: In this article, the authors answer some questions arising from a recent paper of Campbell, Heggie, Robertson and Thomas on one-relator free products of two cyclic groups and show how publicly accessible computer programs can be used to help answer questions about finite group presentations.
Abstract: We answer some questions which arise from a recent paper of Campbell, Heggie, Robertson and Thomas on one-relator free products of two cyclic groups. In the process we show how publicly accessible computer programs can be used to help answer questions about finite group presentations.
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TL;DR: In this paper, Voiculescu's random matrix model for freeness is extended to the non-Gaussian case and also the case of constant block diagonal matrices.
Abstract: Voiculescu's random matrix model for freeness is extended to the non-Gaussian case and also the case of constant block diagonal matrices. Thus we are able to investigate free products of free group factors with matrix algebras and with the hyperfinite II$_1$ factor, showing that $$ L(F_n) * R = L(F_(n+1)) $$ for $n \ge 1$, (where $L(F_1)=L(Z)$).
TL;DR: In this paper, the Cayley graph of the free product of q + 1 copies of Z n+1 is considered as a group acting on the hyperbolic disk, and harmonic analysis on these groups is studied.
Abstract: Let be the free product of q + 1 copies of Z n+1 and let denote its Cayley graph (with respect to a j , 1 ≤ j ≤ q + 1). We may think of G as a group acting on the “homogeneous space” , This point of view is inspired by the case of SL 2 (R) acting on the hyperbolic disk and is developed in [FT-P] [I-P] [FT-S] [S] (but see also [C]). Since G is a group we may investigate some classical topics: the full (reductive) C * algebra, its dual space, the regular Von Neumann algebra and so on. See [B] [P] [L] [V] and also [H]. These approaches give results pointing up the analogy between harmonic analysis on these groups and harmonic analysis on more classical objects.
TL;DR: In this paper, the small cancellation theory over free products with amalgamation and HNN groups is extended to groups acting on trees in which the action with inversions is possible, including tree products of groups and treed-HNN groups.
Abstract: The small cancellation theory over free products with amalgamation and HNN groups is extended to groups acting on trees in which the action with inversions is possible. This will include the case of tree products of groups and treed-HNN groups.
TL;DR: In this paper, the authors considered the class consisting of all groups constructive from cyclic groups using amalgamated free products and HNN-extensions, with certain restrictions, and obtained a description of all the subgroups of groups in that satisfy identities, and showed that the groups in satisfy the Tits alternative.
Abstract: A description is obtained of the subgroups of groups acting on a tree that do not contain nonabelian free subgroups; it is a new interpretation of a result of Bass. The author considers the class consisting of all groups constructive from cyclic groups using amalgamated free products and HNN-extensions, with certain restrictions. A description is obtained of all the subgroups of groups in that satisfy identities, and it is shown that the groups in satisfy the Tits alternative. The proof uses the techniques of group actions on trees.
TL;DR: In this paper, the authors defined the residual complex as a group and its subgroups acting on a complex and proved that it becomes a clear step complex if the group can be expressed as an amalgamated free product of its sub groups.
Abstract: In this paper the “residual complex” is defined when a group and its subgroups act on a complex. With its aid a homological spectral sequence of group products is given. And the author makes a concentrated study of the structure of the residual complex and proves that it becomes a clear “step complex” if the group can be expressed as an amalgamated free product of its subgroups.