Showing papers on "Free product published in 2015"
TL;DR: In this paper, the authors studied Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtained several uniqueness and non-existence results.
Abstract: We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $\Gamma$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ arising from a free ergodic probability measure preserving action of $\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability measure preserving equivalence relations, then the II$_1$ factor $L(\mathcal R)$ has a unique Cartan subalgebra, up to unitary conjugacy. Finally, we show that the free product $M=M_1*M_2$ of any two II$_1$ factors does not have a Cartan subalgebra. More generally, we prove that if $A\subset M$ is a diffuse amenable von Neumann subalgebra and $P\subset M$ denotes the algebra generated by its normalizer, then either $P$ is amenable, or a corner of $P$ embeds into $M_1$ or $M_2$.
57 citations
TL;DR: In this article, the Lipschitz metric for the outer space of a free product of groups was developed and the existence of simplicial train track maps was shown to exist in both free group and free product cases.
Abstract: In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices. In particular, we describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths. We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed. We include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory. A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.
33 citations
TL;DR: In this article, a complete classification of the factors (P, ϕ ) F n ⋊ F n, where F n is the free group and P is an amenable factor with an almost periodic state ϕ is given.
29 citations
TL;DR: In this paper, the authors generalize the graphical small cancellation theory of Gromov to the free product and construct a large number of non-isomorphic torsion-free groups without the unique product property.
29 citations
TL;DR: In this article, it was shown that a class of groups is root in the sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products.
Abstract: We prove that a class of groups is root in a sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we obtain a condition which is sufficient for the generalized free product of two nilpotent groups to be residual solvable.
23 citations
TL;DR: In this paper, the authors introduced the problem of finding optimal time hypercontractivity bounds for the Ornstein-Uhlenbeck semigroup acting on Clifford algebras, and showed how one can extend this result to obtain optimal hyper-contractivity estimates for the free product extension of that semigroup.
Abstract: In this seminar we will introduce the problem of finding optimal time hypercontractivity bounds for the Ornstein-Uhlenbeck semigroup acting on Clifford algebras. Moreover, we will show how one can extend this result to obtain optimal time hypercontractivity estimates for the free product extension of that semigroup. This generalization is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. In particular, this result generalizes the work of Nelson, Gross, Carlen/Lieb and Biane.
20 citations
TL;DR: Berbec and Vaes as mentioned in this paper showed that for weakly amenable groups Γ having positive first l2-Betti number, the same wreath product group 𝒢 is W*superrigid.
Abstract: In [M. Berbec and S. Vaes, W*-superrigidity for group von Neumann algebras of left–right wreath products, Proc. London Math. Soc.108 (2014) 1116–1152] we have proven that, for all hyperbolic groups and for all nontrivial free products Γ, the left–right wreath product group 𝒢 ≔ (ℤ/2ℤ)(Γ) ⋊ (Γ × Γ) is W*-superrigid, in the sense that its group von Neumann algebra L𝒢 completely remembers the group 𝒢. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups Γ having positive first l2-Betti number, the same wreath product group 𝒢 is W*-superrigid.
18 citations
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TL;DR: In this article, it was shown that the set of all solutions in reduced words is an EDT0L language, which is a proper subclass of indexed languages, and that it is an indexed language in the sense of Aho.
Abstract: We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been been open for some years, apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $\star_{1 \leq i \leq s}F_i$, where each $F_i$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by Ferte, Marin and Senizergues (ToCS 2014), which is a very restricted case. Our general result became possible due to the recent recompression technique of Jez. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper (arXiv 1405.5133). For example, we improve the complexity from quadratic nondeterministic space in (arXiv 1405.5133) to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE($n\log n$). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.
17 citations
TL;DR: In this article, necessary and sufficient conditions for the free product of two groups with normal amalgamated subgroups to be a residually C-group were obtained, where C is a root class of groups which must be homomorphically closed in most cases.
Abstract: We obtain both necessary and sufficient conditions for the free product of two groups with normal amalgamated subgroups to be a residually C-group, where C is a root class of groups, which must be homomorphically closed in most cases.
15 citations
TL;DR: In this article, Martin et al. showed that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homomorphism types of the boundaries of its factors.
Abstract: We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups). Primary MSC. 20F65. Secondary MSC. 20E08, 57M07. Alexandre Martin, Fakultat fur Mathematik, Oskar-Morgenstern-Platz 1, 1180 Wien, Austria. E-mail: alexandre.martin@univie.ac.at Phone: +43 1 4277 850766 Jacek Świątkowski, Instytut Matematyczny, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland E-mail: Jacek.Swiatkowski@math.uni.wroc.pl Phone: +48 71 3757491
15 citations
26 Aug 2015
TL;DR: In this paper, it was shown that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set.
Abstract: We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex stabilizers and some more general, finite or infinite, edge stabilizers that we call highly core-free. We study the notion of highly core-free subgroups and give some examples. In the case of a free product amalgamated over a highly core-free subgroup and an HNN extension with a highly core-free base group we obtain a genericity result for faithful and highly transitive actions.
TL;DR: In this article, it was shown that the tame subgroup TA3(K) of the group GA3(k) of polynomial automorphisms of AK can be realized as a generalized amalgamated product.
Abstract: For K a field of characteristic zero, it is shown that the tame subgroup TA3(K) of the group GA3(K) of polynomial automorphisms of AK can be realized as a generalized amalgamated product, specifically, the product of three subgroups, amalgamated along pairwise intersections, in a manner that generalizes the well-known amalgamated free product structure of TA2(K) (which coincides with GA2(K) by Jung’s Theorem). The result follows from defining relations for TA3(K) given by U. U. Umirbaev.
TL;DR: In this article, it was shown that stable commutator length is rational on free products of free abelian groups amalgamated over Ωk, a class of groups containing the fundamental groups of all torus knot complements.
Abstract: We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.
TL;DR: The rank gradient and p-gradient of free products with amalgamation over an amenable subgroup and HNN extensions with anAmenable associated subgroup are calculated.
Abstract: We calculate the rank gradient and p-gradient of free products with amalgamation over an amenable subgroup and HNN extensions with an amenable associated subgroup. The notion of cost is used to compute the rank gradient of amalgamated free products and HNN extensions. For the p-gradient the Kurosh subgroup theorems for amalgamated free products and HNN extensions will be used.
TL;DR: In this paper, it was shown that some one-relator groups, such as Baumslag-Solitar groups, are n-slender, and the same authors also proved a corresponding property for certain HNN extensions and amalgamated free products.
Abstract: In 2011, while investigating fundamental groups of wild spaces, K.Eda [7] showed that the fundamental group of the Hawaiian earring (the Hawaiian earring group, in short) has the property that for any homomorphism h from it to a free product A*B, there exists a natural number N such that is contained in a conjugate subgroup to A or B. In the present article, we prove a corresponding property for certain HNN extensions and amalgamated free products. This allows us to show that some one-relator groups, including Baumslag–Solitar groups, are n-slender.
TL;DR: A Metropolis Monte Carlo algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group is described, which allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability.
Abstract: We describe a Metropolis Monte Carlo algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm samples from a stretched Boltzmann distribution where |w| is the length of a word w, α and β are parameters of the algorithm, and Z is a normalizing constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the Baumslag–Solitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), the genus 2 surface group, and Richard Thompson’s group F.
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TL;DR: In this article, the authors studied the structure of, and relationships between, various subgroups of G defined by the dynamics of H. In particular, they considered the intersection of all tidy subgroups for H on G and the smallest closed H-invariant subgroup D such that H acts distally on G/D.
Abstract: Let G be a totally disconnected, locally compact group and let H be a virtually flat (for example, polycyclic) group of automorphisms of G. We study the structure of, and relationships between, various subgroups of G defined by the dynamics of H. In particular, we consider the following four subgroups: the intersection of all tidy subgroups for H on G (in the case that H is flat); the intersection of all H-invariant open subgroups of G; the smallest closed H-invariant subgroup D such that H acts distally on G/D; and the group generated by the closures of contraction groups of elements of H on G.
TL;DR: For a finitely generated group, the relations between its rank, the maximal rank of its free quotient, called co-rank, and the Betti number are studied in this article.
Abstract: For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group’s rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive with respect to the free product and the direct product of groups. Our results are important for the theory of foliations and for manifold topology, where the corresponding notions are related with the cut-number (or genus) and the isotropy index of the manifold, as well as with the operations of connected sum and direct product of manifolds.
TL;DR: In this article, Ribes and Zalesskii showed that a group G is subgroup conjugacy separable if for every pair of non-conjugate subgroups H and K of G, there exists a finite quotient of G where the images of these subgroups are not conjugate.
Abstract: A group G is said to be conjugacy subgroup separable if for every pair of non-conjugate finitely generated subgroups H and K of G, there exists a finite quotient of G where the images of these subgroups are not conjugate. The notion was introduced recently by O. Bogopolski and F. Grunewald. We prove here that finitely generated free-by-finite groups are subgroup conjugacy separable. This generalizes Theorem 1.5 in Bogopolski and Grunewald (On subgroup conjugacy separability in the class of virtually free groups, vol 110, 18 pages, 2010). We also show that free products preserve subgroup conjugacy separability. The methods are based on the profinite version of Bass–Serre’s theory of groups acting on trees. In particular, we use essentially the results of Ribes and Zalesskii (Rev Mat Iberoam 30:165–190, 2014).
TL;DR: In this article, it was shown that every extension of a free group by a C-group is conjugacy C-separable, where C is an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group.
Abstract: Let C be an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group. It is proved that every extension of a free group by a C-group is conjugacy C-separable. It is also proved that, if G is a free product of two conjugacy C-separable groups with finite amalgamated subgroup or an HNN-extension of a conjugacy C-separable group with finite associated subgroups, then the group G is residually C if and only if it is conjugacy C-separable.
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TL;DR: In this paper, it was shown that a discrete quantum group is torsion-free under Cartesian and free products if and only if its associated fusion ring is Torsion free.
Abstract: Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is preserved under Cartesian and free products. We also discuss torsion-freeness in the more general setting of abstract rigid tensor C*-categories
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TL;DR: In this paper, a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations was proved, and the results unify, simplify and generalize all the previous results obtained before.
Abstract: We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen.
Abstract: Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO(G) of all left-invariant circular orders on a fixed countable and discrete group G. CO(G) contains as a closed subspace LO(G), the space of all left-invariant linear orders of G, as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the action of Aut(G) on CO(G) and relate it to results of Koberda regarding the action on LO(G). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups A, we use a classification of elements of CO(A) to describe the homeomorphism type of the space CO(A), and to show that Aut(A) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.
TL;DR: In this article, the authors interpret several constructions with C*-algebras as colimits in the bicategory of correspondences, including crossed products for actions of groups and crossed modules.
Abstract: We interpret several constructions with C*-algebras as colimits in the bicategory of correspondences. This includes crossed products for actions of groups and crossed modules, Cuntz-Pimsner algebras of proper product systems, direct sums and inductive limits, and certain amalgamated free products.
17 Nov 2015
TL;DR: In this article, a non-commutative generalization of some probabilistic results that occur in representation theory is presented, and the results are divided into three different parts: the first part of the thesis is a classification of unitary easy quantum groups whose intertwiner spaces are described by non-crossing partitions.
Abstract: The subject of this thesis is the non-commutative generalization of some probabilistic results that occur in representation theory. The results of the thesis are divided into three different parts. In the first part of the thesis, we classify all unitary easy quantum groups whose intertwiner spaces are described by non-crossing partitions, and develop the Weingarten calculus on these quantum groups. As an application of the previous work, we recover the results of Diaconis and Shahshahani on the unitary group and extend those results to the free unitary group. In the second part of the thesis, we study the free wreath product. First, we study the free wreath product with the free symmetric group by giving a description of the intertwiner spaces: several probabilistic results are deduced from this description. Then, we relate the intertwiner spaces of a free wreath product with the free product of planar algebras, an object which has been defined by Bisch and Jones. This relation allows us to prove the conjecture of Banica and Bichon. In the last part of the thesis, we prove that the minimal and the Martin boundaries of a graph introduced by Gnedin and Olshanski are the same. In order to prove this, we give some precise estimates on the uniform standard filling of a large ribbon Young diagram. This yields several asymptotic results on the filling of large ribbon Young diagrams
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TL;DR: In this paper, the set of subgroups of the automorphism group of a full shift and submonoids of its endomorphism monoid are discussed and closed under direct products and free products in the group case.
Abstract: We discuss the set of subgroups of the automorphism group of a full shift, and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case, and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.
TL;DR: In this paper, it was shown that G is an almost residually finite π-group if and only if so are A, B, A/H, and B/H.
Abstract: Let G be a free product of almost soluble groups A and B of finite rank with amalgamated normal subgroup H, where H ≠ A and H ≠ B, and let π be a finite set of primes. We prove that G is an almost residually finite π-group if and only if so are A, B, A/H, and B/H.
TL;DR: In this article, it was shown that if a non-cyclic subgroup H of the n-periodic product of a given family of groups is not conjugate to any subgroup of the product's components, then H contains a subgroup isomorphic to the free Burnside group B(2, n).
Abstract: We prove that n-periodic products (introduced by the first author in 1976) are uniquely characterized by certain quite specific properties. Using these properties, we prove that if a non-cyclic subgroup H of the n-periodic product of a given family of groups is not conjugate to any subgroup of the product's components, then H contains a subgroup isomorphic to the free Burnside group B(2, n). This means that H contains the free periodic groups B(m, n) of any rank m > 2, which lie in B(2, n) ([1], Russian p. 26). Moreover, if H is finitely generated, then it is uniformly non-amenable. We also describe all finite subgroups of n-periodic products.
TL;DR: In this paper, the authors use the description of Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini et al. (J. Algebra 285:706-725, 2005) to obtain structural results for such amalgams.
Abstract: We use the description of the Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini et al. (J. Algebra 285:706–725, 2005) to obtain structural results for such amalgams. Schutzenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs that contain all the essential information about the automaton. Using this crucial fact, and Bass–Serre theory, we show that the maximal subgroups of an amalgamated free product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups built from the maximal subgroups of the original semigroups.