Showing papers on "Free product published in 2005"

Hyperbolic Group C -Algebras and Free-Product C -Algebras as Compact Quantum Metric Spaces

Abstract: Let l be a length function on a group G, and let Ml denote the operator of pointwise multi- plication by l on l 2 (G). Following Connes, Ml can be used as a "Dirac" operator for C � r (G). It defines a Lipschitz seminorm on C � r (G), which defines a metric on the state space of C � r (G). We show that if G is a hyperbolic group and if l is a word-length function on G, then the topology from this metric coincides with the weak-∗ topology (our definition of a "compact quantum metric space"). We show that a convenient framework is that of filtered C � -algebras which satisfy a suitable "Haagerup-type" condition. We also use this framework to prove an analogous fact for certain reduced free products of C � -algebras.

Expanders, rank and graphs of groups

Abstract: Let G be a finitely presented group, and let (Gi} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. Gi is an amalgamated free product or HNN extension, for infinitely many i; 2. the Cayley graphs of G/Gi (with respect to a fixed finite set of generators for G) form an expanding family; 3. infi(d(Gi) - 1)/[G : Gi] -- O, where d(Gi) is the rank of Gi. The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.

Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Abstract: Let be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin–Razborov diagrams for . We also prove that every system of equations over is equivalent to a finite subsystem, and a number of structural results about –limit groups.

The fundamental groups of subsets of closed surfaces inject into their first shape groups

Abstract: We show that for every subset X of a closed surface M 2 and every x0 ∈ X, the natural homomorphism ϕ : π1(X, x0) → y π1(X, x0), from the fundamental group to the first shape homotopy group, is injective. In particular, if X ( M 2 is a proper compact subset, then π1(X, x0) is isomorphic to a subgroup of the limit of an inverse sequence of finitely generated free groups; it is therefore locally free, fully residually free and residually finite. AMS Classification 55Q52, 55Q07, 57N05; 20E25, 20E26

Khintchine type inequalities for reduced free products and Applications

Abstract: We prove Khintchine type inequalities for words of a fixed length in a reduced free product of $C^*$-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length $d$ is completely bounded with norm depending linearly on $d$. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema's theorem on the stability of exactness under the reduced free product for $C^*$-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional $C^*$-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak-$*$ CCAP. In the case of group $C^*$-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.

Strongly 1-bounded von Neumann algebras

Abstract: Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $\alpha >0$, F is $\alpha$-bounded if the free packing $\alpha$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x in F with finite free entropy. It is shown that if M is strongly 1-bounded, then any finite set of selfadjoint generators G for M is 1-bounded and the microstates free entropy dimension of G is less than or equal to 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and the microstates free entropy dimension is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II_1-factors which have property Gamma, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of SL_n(Z), n >2. If M and N are strongly 1-bounded and their intersection is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II_1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.

Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups

Abstract: From a group H and h e H, we define a representation ρ: B n → Aut(H* n ), where B n denotes the braid group on n strands, and H* n denotes the free product of n copies of H. We call p the Artin type representation associated to the pair (H, h). Here we study various aspects of such representations. Firstly, we associate to each braid β a group Γ (H,h) (β) and prove that the operator Γ (H,h) determines a group invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant Γ (H,h) , and we prove that the Artin type representations are faithful if and only if h is nontrivial. The last part of the paper is devoted to the study of some semidirect products H* n × ρ B n , where ρ: B n → Aut(H* n ) is an Artin type representation. In particular, we show that H* n × ρ B n is a Garside group if H is a Garside group and h is a Garside element of H.

Random Walks on Groups and Monoids with a Markovian Harmonic Measure

Abstract: We consider a transient nearest neighbor random walk on a group $G$ with finite set of generators $S$. The pair $(G,S)$ is assumed to admit a natural notion of normal form words where only the last letter is modified by multiplication by a generator. The basic examples are the free products of a finitely generated free group and a finite family of finite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a finite set of polynomial equations of degree two. This enables to efficiently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.

The free product of matroids

Abstract: We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh's 1969 conjecture, which provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set.

Residual finiteness of outer automorphism groups of finitely generated non-triangle fuchsian groups

Abstract: In [5] Grossman showed that outer automorphism groups of free groups and of fundamental groups of compact orientable surfaces are residually finite. In this paper we introduce the concept of "Property E" of groups and show that certain generalized free products and HNN extensions have this property. We deduce that the outer automorphism groups of finitely generated non-triangle Fuchsian groups are residually finite.

The Grushko decomposition of a finite graph of finite rank free groups: an algorithm

Abstract: A finitely generated group admits a decomposition, called its Grushko decomposition, into a free product of freely indecomposable groups. There is an algorithm to construct the Grushko decomposition of a finite graph of finite rank free groups. In particular, it is possible to decide if such a group is free.

Irreducible Algebraic Sets in Metabelian Groups

Abstract: We present the construction for a u-product G1 ○ G2 of two u-groups G1 and G2, and prove that G1 ○ G2 is also a u-group and that every u-group, which contains G1 and G2 as subgroups and is generated by these, is a homomorphic image of G1 ○ G2 It is stated that if G is a u-group then the coordinate group of an affine space Gn is equal to G ○ Fn, where Fn is a free metabelian group of rank n Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks

Groups with non-simply connected asymptotic cones

Abstract: We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.

Free-by-Demushkin pro-p groups

Abstract: We give an example of a short exact sequence 1→N→G→D→1 of pro-p groups such that the cohomological dimension cd(G)=2, G is (topologically) finitely generated, N is a free pro-p group of infinite rank, D is a Demushkin group, for every closed subgroup S of G containing N and any natural number n the inflation map Open image in new window is an isomorphism but G is not a free pro-p product of a free pro-p group by a Demushkin group. This is a group theoretic version of a question raised by T. Wurfel for some special Galois groups.

An indecomposable $\bm{PD_3}$-complex whose fundamental group has infinitely many ends

Abstract: We show that $S_3*_{Z/2Z}S_3$ satisfies Turaev's criterion to be the fundamental group of an orientable $PD_3$ -complex. Since this group has infinitely many ends but is indecomposable as a free product this provides a negative answer to a question of Wall.

Attainability of the Exponent of Exponential Growth in Free Products of Cyclic Groups

Abstract: In the paper, the set of exponents of exponential growth (growth exponents) for a finitely generated group with respect to all possible generators of this group is studied. It is proved that the greatest lower bound of this set is attained for the free products of a cyclic group of prime order and a free group of finite rank.

Groups with the weak minimal condition for non-subnormal subgroups II

Abstract: Let G be a group with the property that there are no infinite descending chains of non-subnormal subgroups of G for which all successive indices are infinite. The main result is that if G is a locally (soluble-by-finite) group with this property then either G has all subgroups subnormal or G is a soluble-by-finite minimax group. This result fills a gap left in an earlier paper by the same authors on groups with the stated property.

On the residual solvability of generalized free products of finitely generated nilpotent groups

Abstract: In this paper we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these kinds of structures are often residually solvable.

Non-Hopfian Relatively Free Groups

Abstract: To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960’s, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-Hopfian.

Algebraic properties of semi-simple lattices and related groups.

Abstract: Two abstract theories are developed. The first concerns isomorphism in variants with the same multiplicative properties as the Euler characteristic. It is used to show that the index of a subgroup in a semi-simple lattice is deter mined by its isomorphism type when that index is finite. This is also proved to be the case for subgroups of finite index in free products of finitely many semi-simple lattices as well as certain non-trivial extensions of Z by surface groups. In addition, a criterion for the failure of this property is given which applies to a large class of central extensions. The second development concerns the syzygies of groups. The results of this theory are used to define the cohomology groups of a duality group in terms of morphisms between stable modules in the derived category. The Farrell cohomology of virtual duality groups is also considered.

Property (FA) and lattices in SU(2,1)

Abstract: In this paper we consider Property (FA) for lattices in SU(2,1). First, we prove that SU(2,1;O_3) has Property (FA). We then prove that the arithmetic lattices in SU(2,1) of second type arising from congruence subgroups studied by Rapoport--Zink and Rogawski cannot split as a nontrivial free product with amalgamation; one such example is Mumford's fake projective plane. In fact, we prove that the fundamental group of any fake projective plane has Property (FA).

The outer space of a free product

Abstract: We associate a contractible ``outer space'' to any free product of groups G=G_1*...*G_q. It equals Culler-Vogtmann space when G is free, McCullough-Miller space when no G_i is Z. Our proof of contractibility (given when G is not free) is based on Skora's idea of deforming morphisms between trees. Using the action of Out(G) on this space, we show that Out(G) has finite virtual cohomological dimension, or is VFL (it has a finite index subgroup with a finite classifying space), if the groups G_i and Out(G_i) have similar properties. We deduce that Out(G) is VFL if G is a torsion-free hyperbolic group, or a limit group (finitely generated fully residually free group).

Proper Actions of Automorphism Groups of Free Products of Finite Groups

Abstract: If G is a free product of finite groups, let ΣAut1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutierrez–Krstic derived (also see Bogley–Krstic) space of pointed trees is an $\underline{E} \Sigma {\rm Aut}_1(G)$-space for these groups.

Equivariant Extensions of Categorical Groups

Abstract: If Γ is a group, then the category of Γ-graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from Γ. In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreier’s theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).

On the growth of groups and automorphisms

Abstract: We consider the growth functions βΓ(n) of amalgamated free products Γ = A *C B, where A ≅ B are finitely generated, C is free abelian and |A/C| = |A/B| = 2 For every d ∈ ℕ there exist examples with βΓ(n) ≃ nd+1βA(n) There also exist examples with βΓ(n) ≃ en Similar behavior is exhibited among Dehn functions