Showing papers on "Free product published in 2016"
TL;DR: In this paper, a countable descending chain of easy quantum groups interpolating between Bichon's free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product was shown.
Abstract: We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.
65 citations
TL;DR: It is shown that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it.
40 citations
TL;DR: In this article, it was shown that the free product von Neumann algebra retains the cardinality and each nonamenable factor up to stably inner conjugacy, after permutation of the indices.
Abstract: Let be any nonempty set and let be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class of (possibly type ) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product , we show that the free product von Neumann algebra retains the cardinality and each nonamenable factor up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type factors and is new for free product type factors. It moreover provides new rigidity phenomena for type factors.
34 citations
TL;DR: In this article, it was shown that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel, and that the transitivity degree of an infinite group can only take two values, namely 1 and ∞.
Abstract: We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here, by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.
31 citations
TL;DR: In this paper, the fusion rules of the free wreath product quantum groups G≀⁎SN+ for all compact matrix quantum groups of Kac type G and N≥4.
26 citations
TL;DR: In this paper, the authors studied analogues of classical Hilbert transforms as fourier multipliers on free groups and proved their complete boundedness on non commutative spaces associated with the free group von Neumann algebras for all $1
Abstract: We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1
21 citations
TL;DR: In this article, the root-class residuality of a generalized free product of two residually K -groups with amalgamated subgroups with a retract of one of the factors is established.
Abstract: Given a class K of groups, we prove that the free product of a K -group A and a residually K -group B with amalgamated subgroup which is a retract of B is a residually K -group. We also obtain a sufficient condition for the root-class residuality of a generalized free product of two residually K -groups with amalgamated subgroup which is a retract of one of the factors.
15 citations
Posted Content•
TL;DR: In this article, a new proof of Duncan-Howie's theorem was given, which generalizes and generalizes the theorem of Duncan and Howie, showing that any element not conjugate into a free product of torsion-free groups can be any element in the free product.
Abstract: Let $G=*_\lambda G_\lambda$ be a free product of torsion-free groups, and let $g\in[G,G]$ be any element not conjugate into a $G_\lambda$. Then scl$_G(g)\ge1/2$. This generalizes, and gives a new proof of a theorem of Duncan-Howie.
15 citations
TL;DR: In this article, a new Hopf algebra is defined, called the graded twisting of A, which is a twist of A by a pseudo-2-cocycle, where the action is by adjoint maps.
Abstract: Given a Hopf algebra A graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of A. If the action is by adjoint maps, this new Hopf algebra is a twist of A by a pseudo-2-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and q-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the Kazhdan–Wenzl categories in the general semisimple case cannot be always realized as representation categories of compact quantum groups, and for genuine compact groups, we analyze quantum subgroups of the new twisted compact quantum groups, providing a full description when the twisting group is cyclic of prime order.
11 citations
TL;DR: In this article, isolated left orderings of groups whose positive cones are finitely generated are constructed using an amalgamated free product of two groups having an isolated ordering, and they construct a lot of new examples of isolated orderings, and give an example of isolated left ordering with various properties which previously known isolated ordering does not have.
Abstract: We give a new method to construct isolated left orderings of groups whose positive cones are finitely generated. Our construction uses an amalgamated free product of two groups having an isolated ordering. We construct a lot of new examples of isolated orderings, and give an example of isolated left orderings with various properties which previously known isolated orderings do not have.
10 citations
Posted Content•
TL;DR: The first known infinite non-abelian strongly real strongly real Beauville groups for any odd prime was given in this article by considering the lower central quotients of the free product of two cyclic groups.
Abstract: We give an infinite family of non-abelian strongly real Beauville $p$-groups for any odd prime $p$ by considering the lower central quotients of the free product of two cyclic groups of order $p$. This is the first known infinite family of non-abelian strongly real Beauville $p$-groups.
TL;DR: In this paper, it was shown that the Connes embedding problem is stable under graph products, i.e., a graph product of separable II 1-factors that each embed into R U embeds again into RU, i. e.
Abstract: We prove that a graph product of separable II1-factors that each embed into R U embeds again into R U , i.e. the Connes embedding problem is stable under graph products. Graph products from a group theoretical construction generalizing free products by adding commutation relations that are dictated by a graph. The construction was first considered by Green in her thesis (Gr90) and important examples of graph products arise as right angled Coxeter groups and right angled Artin groups. The formal definition is as follows. Definition 0.1. Let be a graph with vertex set V and edge set E. We may assume that has no double edges and no loops, i.e. (v,v) 6∈E. For v ∈ V let Gv be discrete group. Let G be the graph product group which is the discrete group freely generated by Gv,v ∈ V subject to the relation sts −1 t −1 = 1 whenever s ∈ Gv and t ∈ Gw with (v,w) ∈ E.
TL;DR: In this article, it was shown that for any type III 1 free product factor, its continuous core is full if and only if its τ -invariant is the usual topology on the real line.
Abstract: We prove that, for any type III 1 free product factor, its continuous core is full if and only if its τ -invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free Araki–Woods factors. Moreover, our method shows the same result for full (generalized) Bernoulli crossed product factors of type III 1 .
Posted Content•
TL;DR: For a subgroup of a free product of finite groups, this paper obtained necessary conditions (on its Kurosh decomposition) to be verbally closed for the subgroup to be closed.
Abstract: For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) to be verbally closed.
TL;DR: In this article, the authors give sufficient conditions ensuring that a free product of residually amenable groups is again residually and#x1d49e;, and analogous conditions are given for LE-and locally embeddable into amenable (LEA) groups.
Abstract: Let 𝒞 be a class of groups. We give sufficient conditions ensuring that a free product of residually 𝒞 groups is again residually 𝒞, and analogous conditions are given for LE-𝒞 groups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products.Moreover, we consider the pro-𝒞 topology and we characterize special HNN extensions and amalgamated free products that are residually 𝒞, where 𝒞 is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.
Posted Content•
TL;DR: This article showed that there is no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language.
Abstract: We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristobal Rivas stating that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup.
TL;DR: In this paper, it was shown that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces, and that a group has asymPTP C if and only if
Abstract: We show that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if $G$ and $H$ are groups with asymptotic property C, then both $G \times H$ and $G * H$ have asymptotic property C. We also prove that a group~$G$ has asymptotic property C if $1\to K\to G\to H\to 1$ is exact, if $\operatorname{asdim} K<\infty$, and if $H$ has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk, of Bell and Moran, and an open problem in topology from the Lviv Topological Seminar.
TL;DR: In this article, a synthesis of Bass-Serre theory, preceded by a survey on Cayley graphs and graphs of groups, is presented, and the results on constructibility of a torsion-free hyperbolic group from the algebraic closure of a subgroup are given.
Abstract: In Chapter 1 we give basics on combinatorial group theory, starting from free groups and proceeding with the fundamental constructions: free products, amalgamated free products and HNN extensions. We outline a synthesis of Bass-Serre theory, preceded by a survey on Cayley graphs and graphs of groups. After proving the main theorem of Bass-Serre theory, we present its application to the proof of Kurosh subgroup theorem. Subsequently we recall main definitions and properties of hyperbolic spaces. In Section 1.4 we define algebraic and definable closures and recall a few other notions of model theory related to saturation and homogeneity. The last section of Chapter 1 is devoted to asymptotic cones. In Chapter 2 we prove a theorem similar to Bestvina-Paulin theorem on the limit of a sequence of actions on hyperbolic graphs. Our setting is more general: we consider Bowditch-acylindrical actions on arbitrary hyperbolic graphs. We prove that edge stabilizers are (finite bounded)-by-abelian, that tripod stabilizers are finite bounded and that unstable edge stabilizers are finite bounded. In Chapter 3 we introduce the essential notions on limit groups, shortening argument and JSJ decompositions. In Chapter 4 we present the results on constructibility of a torsion-free hyperbolic group from the algebraic closure of a subgroup. Also we discuss constructibility of a free group from the existential algebraic closure of a subgroup. We obtain a bound to the rank of the algebraic and definable closures of subgroups in torsion-free hyperbolic groups. In Section 4.2 we prove some results about the position of algebraic closures in JSJ decompositions of torsion-free hyperbolic groups and other results for free groups. Finally, in Chapter 5 we answer the question about equality between algebraic and definable closure in a free group. A positive answer has been given for a free group F of rank smaller than 3. Instead, for free groups of rank strictly greater than 3 we found some counterexample. For the free group of rank 3 we found a necessary condition on the form of a possible counterexample.
TL;DR: In this paper, it was shown that being fully residually free is equivalent to being commutative transitive (CT), which is also equivalent to having the same universal theory as the class of nonabelian free groups.
Abstract: Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman and Remeslennikov showed that this is also equivalent to being universally free, that is, having the same universal theory as the class of nonabelian free groups. This result is one of the cornerstones of the proof of the Tarski problems. In this article, we provide new examples of groups for which Benjamin Baumslag's theorem is true, that is, we consider classes of groups 𝒳 for which a group is fully residually 𝒳 if and only if it is residually 𝒳 and commutative transitive.We show that this is true for many important classes of groups, including those of free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups (done by Kharlamapovich and Myasnikov), and one-relator groups with only odd torsion. Furthermore, we show that many of the properties discussed here are closed under taking free products. We then consider ...
Posted Content•
TL;DR: The Hochschild and Gerstenhaber-Schack cohomological dimensions of universal cosovereign Hopf algebras were derived in this article, when the matrix of parameters is a generic asymmetry.
Abstract: We compute the Hochschild and Gerstenhaber-Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.
TL;DR: This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.
Abstract: In this paper, we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free σ-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.
TL;DR: In this paper, it was shown that any reduced amalgamated free product C*-algebra is KK-equivalent to the corresponding full amalgamate free product (F * ) C* algebra, and the main ingredient of its proof is Julg-Valette's geometric construction of Fredholm modules with Connes's view for representation theory of operator algebras.
Abstract: We prove that any reduced amalgamated free product C*-algebra is KK-equivalent to the corresponding full amalgamated free product C*-algebra. The main ingredient of its proof is Julg--Valette's geometric construction of Fredholm modules with Connes's view for representation theory of operator algebras.
[...]
TL;DR: In this article, the stable commutator length (scl) in free products via surface maps into a wedge of spaces was studied and it was shown that scl is piecewise rational linear if it vanishes on each factor of the free product.
Abstract: We study stable commutator length (scl) in free products via surface maps into a wedge of spaces. We prove that scl is piecewise rational linear if it vanishes on each factor of the free product, generalizing the main result in Danny Calegari's paper "Scl, sails and surgery". We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show independently and in a new approach that scl in free products of cyclic groups behaves in a piecewise quasi-rational way when the word is fixed but the orders of factors vary, previously proved by Timothy Susse, settling a conjecture of Alden Walker.
01 Jan 2016
TL;DR: In this paper, the authors studied the geodesic growth of finitely generated groups and gave lower bounds for the minimal growth rate of a free product of the form C 2 ∗ Cn.
Abstract: The objective of this Thesis is to study the geodesic growth of finitely generated groups. Firstly, we study direct, free and wreath products of groups. More specifically, we give lower bounds for the minimal geodesic growth rates of abelian groups and upper bounds for the minimal geodesic growth rates of direct products of two groups. Moreover, we give the minimal geodesic growth rate of a free product of the form C2 ∗ Cn, a lower bound for the geodesic growth rate of a free product of two groups, with respect to the standard generating set, and prove that every non trivial free product whose minimal geodesic growth rate is achieved is Hopfian. Also, we study the geodesic growth rate of Lamplighter groups and give the geodesic growth rates of L2 and L3 with respect to the standard generating set. Secondly, we study the geodesic growth rate of some groups acting on regular rooted trees, groups which were known or conjectured to have intermediate spherical growth. We prove, using Schreier graphs, that almost all of these groups have exponential geodesic growth. The exception is the Gupta-Fabrykowski group, for which we show that it is not feasible to prove that the geodesic growth is exponential using Schreier graphs. Finally, we study the rationality of geodesic growth series for graph products and wreath products. We prove that the free product and direct product of two groups of rational geodesic growth have rational geodesic growth with respect to the standard generating sets. Afterwards we prove that the wreath product A oG, where A has rational geodesic growth and G is finite and acts on A, has rational geodesic growth, and that the Lamplighter groups L2 and L3 have rational geodesic growth. Finally, we give an example of a group which has the h-FFTP property and a non-context-free geodesic language.
TL;DR: In this article, the authors generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products and obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a free product of Guirardel--Levitt.
Abstract: We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a free product of Guirardel--Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT($0$). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained.
TL;DR: In this article, the authors examined the completely isometric automorphisms of a free product of noncommutative disc algebras and showed that such an automorphism can be obtained by a permutation of the components of the free product.
Abstract: We examine the completely isometric automorphisms of a free product of noncommutative disc algebras. It will be established that such an automorphism is given simply by a completely isometric automorphism of each component of the free product and a permutation of the components. This mirrors a similar fact in topology concerning biholomorphic automorphisms of product spaces with nice boundaries due to Rudin, Ligocka and Tsyganov.
This paper is also a study of multivariable dynamical systems by their semicrossed product algebras. A new form of dynamical system conjugacy is introduced and is shown to completely characterize the semicrossed product algebra. This is proven by using the rigidity of free product automorphisms established in the first part of the paper.
Lastly, a representation theory is developed to determine when the semicrossed product algebra and the tensor algebra of a dynamical system are completely isometrically isomorphic.
TL;DR: The free product of M-fuzzifying matroids is defined when M is a finite Boolean algebra and a number of fundamental properties of the operation are derived.
Abstract: In this paper, motivated by the free product of crisp matroids, the free product of M-fuzzifying matroids is defined when M is a finite Boolean algebra and a number of fundamental properties of the operation are derived. Parallel to crisp matroids, the minors of M-fuzzifying matroids are described in terms of the M-fuzzifying truncation operator and its dual, the M-fuzzifying Higgs lift operator.
TL;DR: In this paper, a general description of the discrete decomposition of type III factors arising as central summands of free product von Neumann algebras is given, and several precise structural results on type III free product factors are given.
Abstract: We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.
TL;DR: In this paper, it was shown that there are infinite sharply 2-transitive groups which do not arise from fields or near-fields, and it is not hard to construct concrete examples not arising from fields.
Abstract: The finite sharply 2-transitive groups were classified by Zassenhaus in the 1930’s. They essentially all look like the group of affine linear transformations $x\mapsto ax+b$
for some field (or at least near-field) $K$
. However, the question remained open whether the same is true for infinite sharply 2-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus’ results might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal structures it was indeed proved that all examples inside the given class arise in this way as affine groups. However, it recently turned out that the reason for the lack of a general proof was the fact that there are plenty of sharply 2-transitive groups which do not arise from fields or near-fields! In fact, it is not too hard to construct concrete examples (see below). In this note, we survey general sharply $n$
-transitive groups and describe how to construct examples not arising from fields.
TL;DR: In this paper, it was shown that the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroups H is residually Fπ-separable.
Abstract: Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually Fπ, where Fπ is the class of all finite p-groups. We prove that G is residually Fπ if and only if H is Fπ-separable in A and B.