Showing papers on "Free product published in 2012"

Cartan subalgebras of amalgamated free product II$_1$ factors

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 $M_2$.

W*-superrigidity for group von Neumann algebras of left-right wreath products

Abstract: We prove that for many nonamenable groups \Gamma, including all hyperbolic groups and all nontrivial free products, the left-right wreath product group G := (Z/2Z)^(\Gamma) \rtimes (\Gamma \times \Gamma) is W*-superrigid. This means that the group von Neumann algebra LG entirely remembers G. More precisely, if LG is isomorphic with L\Lambda for an arbitrary countable group \Lambda, then \Lambda must be isomorphic with G.

Splittings and automorphisms of relatively hyperbolic groups

Abstract: We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P to G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P to G) is infinite, then P is a vertex group in a splitting of G. If P is torsion-free, then Out(P to G) is of type VF, in particular finitely presented. We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitely-ended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.

Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting

Abstract: Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.

Ordering graph products of groups

Abstract: It is shown that a graph product of right-orderable groups is right orderable, and that a graph product of (two-sided) orderable groups is orderable. The latter result makes use of a new way of ordering free products of groups.

A note on unital full amalgamated free products of RFD C∗-algebras

Abstract: In the paper, we consider the question whether a unital full amalgamated free product of RFD (residually finite dimensional) C∗-algebras is RFD again. One example shows that the answer to the general case is no. We give a necessary and sufficient condition such that a unital full amalgamated free product of RFD C∗-algebras with amalgamation over a finite dimensional C∗-algebra is RFD. Applying this result, we conclude that a unital full free product of two same RFD C∗-algebras with amalgamation over a finite-dimensional C∗-algebra is always RFD.

A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups

Abstract: We present a link between easy quantum groups, discrete groups and combinatorics. By this, we infer new connections between quantum isometry groups, reflection groups, varieties of groups and the combinatorics of partitions. More precisely, we consider easy quantum groups and find a relation to subgroups of the infinite free product $\mathbb Z_2^{*\infty}$ of $\mathbb Z_2=\mathbb Z/2\mathbb Z$. We obtain a link with reflection groups and thus with varieties of groups, which yields a statement on the complexity of the class of easy quantum groups on the one hand, and a "quantum invariant" for varieties of groups on the other hand. Moreover, we reveal a triangular relationship between easy quantum groups, categories of partitions and discrete groups (reflection groups). As a by-product, we obtain a large number of new quantum isometry groups.

Free product von Neumann algebras associated to graphs and Guionnet, Jones, Shlyakhtenko subfactors in infinite Depth

Abstract: Given a subfactor planar algebra P, Guionnet, Jones and Shlyakhtenko give a diagrammatic construction of a II_{1} subfactor whose planar algebra is P. They showed if P is finite-depth, then the factors are interpolated free group factors, and they identified the parameters. We prove if P is infinite-depth, then the factors are isomorphic to L(F\infty}$.

Surface subgroups from linear programming

Abstract: We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank 2 free group sending a to ab and b to ba; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension 2) of a double of a free group along a collection of subgroups is a finite-sided rational polyhedron, and that every rational class is virtually represented by an extremal surface subgroup. These results are obtained by a mixture of combinatorial, geometric, and linear programming techniques.

Conjugacy growth series and languages in groups

Abstract: In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.

Classes of Groups Generalizing a Theorem of Benjamin Baumslag

Abstract: In [BB] Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman [GS] and Remeslennikov [Re] 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 Tarksi problems. In this paper we extend the class of groups for which Benjamin Baumslag's theorem is true, that is we consider classes of groups $\X$ for which being fully residually $\X$ is equivalent to being residually $\X$ and commutative transitive. We show that the classes of groups for which this is true is quite extensive and includes free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups (done in [KhM]), and one-relator groups with only odd torsion. Further, the class of groups having this property is closed under certain amalgam constructions, including free products and free products with malnormal amalgamated subgroups. We also consider extensions of these classes to classes where the equivalence with universally $\X$ groups is maintained.

Hyperbolic towers and independent generic sets in the theory of free groups

Abstract: We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a finitely generated model which omits $p_0^{(2)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.

Amalgamated products of groups: measures of random normal forms

Abstract: Let \( G = A\mathop { * }\limits_C B \) be an amalgamated product of finite rank free groups A, B, and C. We introduce atomic measures and corresponding asymptotic densities on a set of normal forms of elements in G. We also define two strata of normal forms: the first one consists of regular (or stable) normal forms, and the second stratum is formed by singular (or unstable) normal forms. In a series of previous works about classical algorithmic problems, it was shown that standard algorithms work fast on elements of the first stratum and nothing is known about their work on the second stratum. In this paper, we give probabilistic and asymptotic estimates of these strata.

Discrete cores of type III free product factors

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.

Ampleness in the free group

Abstract: We show that the theory of the free group – and more generally the theory of any torsionfree hyperbolic group – is n-ample for any n ≥ 1. We give also an explicit description of the imaginary algebraic closure in free groups.

The symmetries of McCullough-Miller space

Abstract: We prove that if W is the free product of at least four groups of order 2, then the automorphism group of the McCullough-Miller space corresponding to W is isomorphic to group of outer automorphisms of W. We also prove that, for each integer n ≥ 3, the automorphism group of the hypertree complex of rank n is isomorphic to the symmetric group of rank n.

A Geometric Proof of the Structure Theorem for Cyclic Splittings of Free Groups

Abstract: We give a geometric proof of a well known theorem that describes splittings of a free group as an amalgamated product or HNN extension over the integers. The argument generalizes to give a similar description of splittings of a virtually free group over a virtually cyclic group.

Exactness of universal free products of finite dimensional C^*-algebras with amalgamation

Abstract: We investigate free products of finite dimensional C∗ -algebras with amalgamation over diagonal subalgebras. We look to determine under what circumstances such a free product is exact and/or nuclear. We completely characterize exactness of Mn ∗D Mk where D is a unital subalgebra of both Mn and Mk . Our characterization depends both on the dimension of D and the embeddings of D into Mj and Mk . We also show that for free products of three finite dimensional algebras exactness fails. Lastly we look at some nonunital embeddings of a diagonal subalgebra into finite dimensional algebras. Mathematics subject classification (2010): 46L09.

Cyclic rewriting and conjugacy problems

Abstract: Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.

Surface Maps into Free Groups

Abstract: We exploit the combinatorial properties of surface maps into free groups to prove several new results in the field of stable commutator length and bounded cohomology. We show that random homomorphisms between free groups are isometries of scl; we prove interesting properties of the scl unit ball; we describe a transfer construction for quasimorphisms and give an infinite family of chains whose scl it certifies; we linearize the dynamics of endomorphisms on free groups and use this to prove that random endomorphisms can be realized by surface immersions, which provides many examples of surface subgroups of HNN extensions of free groups; and finally, we give an algorithm to compute scl in free products of finite or infinite cyclic groups that generalizes and improves previous work.

Class-preserving automorphisms of generalized free products amalgamating a cyclic normal subgroup

Abstract: In general, a class-preserving automorphism of generalized free products of nilpotent groups, amalgamating a cyclic normal subgroup of order 8, need not be an inner automorphism. We prove that every class- preserving automorphism of generalized free products of nitely generated nilpotent groups, amalgamating a cyclic normal subgroup of order less than 8, is inner.

Hypercontractivity for free products

Abstract: In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbeck semigroup acting on the Clifford algebra. Our approach is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. With another use of Speicher's central limit theorem, we may also obtain the same bounds for free products of q-deformed von Neumann algebras interpolating between the fermonic and bosonic frameworks. This generalizes the work of Nelson, Gross, Carlen/Lieb and Biane. Our main application yields hypercontractivity bounds for the free Poisson semigroup acting on the group algebra of the free group Fn, uniformly in the number of generators.

Fixed points of endomorphisms of certain free products

Abstract: The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.

Operator systems from discrete groups

Abstract: We formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group on $n$ generators, as well as the operator systems of the free products of finitely many copies of the two-element group $\mathbb Z_2$. We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes.

Exponential growth rates of free and amalgamated products

Abstract: We prove that there is a gap between $\sqrt{2}$ and $(1+\sqrt{5})/2$ for the exponential growth rate of free products $G=A*B$ not isomorphic to the infinite dihedral group. For amalgamated products $G=A*_C B$ with $([A:C]-1)([B:C]-1)\geq2$, we show that lower exponential growth rate than $\sqrt{2}$ can be achieved by proving that the exponential growth rate of the amalgamated product $\mathrm{PGL}(2,\mathbb{Z})\cong (C_2\times C_2) *_{C_2} D_6$ is equal to the unique positive root of the polynomial $z^3-z-1$. This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011) 208--217].

Distributions of traffics and their free product: an asymptotic freeness theorem for random matrices and a central limit theorem

Abstract: The distributions of traffics are defined and are applied for families of larges random matrices, random groups and infinite random rooted graphs with uniformly bounded degree. There are constructed by adding axioms in Voiculescu's definition of $^*$-distribution of non commutative random variables. The convergence in distribution of traffics generalizes Benjamini, Schramm, Aldous, Lyons' weak local convergence of random graphs. We introduce a notion of freeness of traffics, which contains both the classical notion of independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem for families of matrices invariant by permutation, which enlarges the class of large random matrices for which we can predict the empirical eigenvalues distribution. We prove a central limit theorem for the sum of free traffics, and interpret the limit as the (traffic)-convolution of a gaussian commutative random variable and a semicircular non commutative random variable. We make a connection between the freeness of traffics and the natural free product of random graphs, combination of the statistical independence and of the geometric free product.