Showing papers on "Free product published in 2003"

Constructions preserving Hilbert space uniform embeddability of discrete groups

Abstract: Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.

On sofic groups

Abstract: Answering some queries of Weiss, we prove that the free product and amenable extensions of sofic groups are sofic as well, and give an example of a finitely generated sofic group that is not residually amenable.

Limit groups and groups acting freely on R^n-trees

Abstract: We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R^n-trees. We first prove that Sela's limit groups do have a free action on an R^n-tree. We then prove that a finitely generated group having a free action on an R^n-tree can be obtained from free abelian groups and surface groups by a finite sequence of free products and amalgamations over cyclic groups. As a corollary, such a group is finitely presented, has a finite classifying space, its abelian subgroups are finitely generated and contains only finitely many conjugacy classes of non-cyclic maximal abelian subgroups.

Regular neighbourhoods and canonical decompositions for groups

Abstract: We find canonical decompositions for finitely presented groups which essentially specialise to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood of a family of almost invariant subsets of a group. An almost invariant set is an analogue of an immersion. This article is an announcement of work whose details will appear later. In this work, we study analogues for groups of the classical JSJ-decomposition (see Jaco and Shalen [9], Johannson [10] and Waldhausen [19]) for orientable Haken 3-manifolds. The orientability restriction is not essential but it will simplify our discussions. Previous work in this field has been carried out by Kropholler [11], Sela [18], Rips and Sela [13], Bowditch [2], [3], Dunwoody and Sageev [5], and Fujiwara and Papasoglu [8], but none of these results yields the classical JSJdecomposition when restricted to the fundamental group of an orientable Haken manifold. In our work, we give a new approach to this subject, and we give decompositions for finitely presented groups which essentially specialise to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. An important feature of our approach is that the decompositions we obtain are unique and are invariant under automorphisms of the group. In previous work such strong uniqueness results were only found for decompositions of word hyperbolic groups. Most of the results of the previous authors for virtually polycyclic groups can be deduced from our work. But our arguments use some of the results of these authors, particularly those of Bowditch. In addition, we use the important work of Dunwoody and Roller in [4]. Our work also yields some extensions of the results on the Algebraic Annulus and Torus Theorems in [16], [2], and [6]. It should be remarked that even though we obtain canonical decompositions for all finitely presented groups, these decompositions are often trivial. This is analogous to the fact that any finitely generated group possesses a free product decomposition, but this decomposition is trivial whenever the given group is freely indecomposable. Our ideas are based on an algebraic generalisation of the Enclosing Property of the classical JSJ-decomposition. This property can be described briefly as follows. Received by the editors May 1, 2002, and, in revised form, July 23, 2002. 2000 Mathematics Subject Classification. Primary 20E34; Secondary 57N10, 57M07.

Endomorphisms of kleinian groups

Abstract: Ag roupG is cohopfian (or has the co-Hopf property) if any injective endomorphism f : G → G is surjective. Answering a question of E. Rips, Z. Sela showed in [S2] that a torsionfree, non-virtually cyclic word-hyperbolic group (in Gromov’s sense) is cohopfian if and only if it is not a non-trivial free product. The cohopficity of 3-manifold groups has been studied by many authors; see [PW] and [OP] where a more complete list of references on this subject is given. A non-trivial free product A ∗ B is never cohopfian, as it contains the proper subgroup A∗mBm −1 isomorphic to A∗B if m/ ∈ (A∪B). More generally, let the group G split as an HNN-extension, G = A∗C = � A, t | tCt −1 = ϕ(C)� , and suppose that t centralizes C .T henG is not cohopfian (set f : G → G be the identity on A and f (t )= t 2 ;t henf is injective, not surjective). It is shown in [OP] that this example can be realized as a Kleinian group. Note that in this case, the group G splits over a parabolic subgroup C which is of infinite index in the unique maximal parabolic subgroup ˜

Self-Normalization of Free Subgroups in the Free Burnside Groups

Abstract: Let G be a free Burnside group of sufficiently large exponent n. Assume that a non-trivial subgroup N of G is itself a free Burnside group of exponent n. We prove that N must coincide with its normalizer in G. In particular, if N is a normal free Burnside subgroup of G,then either N = G or N = 1.

Nielsen Methods and Groups Acting on Hyperbolic Spaces

Abstract: We show that for any n & in ℕ there exists a constant C(n) such that any n-generated group G which acts by isometries on a δ-hyperbolic space (with δ>0) is either free or has a nontrivial element with translation length at most δC(n)

Strong marked isospectrality of affine Lorentzian groups

Abstract: The Margulis invariant is a function defined on a group of Lorentzian transformations $G$ acting on Minkowski space $\R^{2,1}$, that contains no elliptic elements. The spectrum of $G$ is the sequence of values of the Margulis invariant for all its elements. If the underlying linear group of $G$ is fixed, Drumm and Goldman proved that the spectrum defines the translational part completely. In this note, we strengthen this result by showing that isospectrality holds for any free product of cyclic groups of given rank, up to conjugation in the group of affine transformations of $R^{2,1}$, as long as it is non-radiant and that its linear part is discrete and non-elementary. In particular, isospectrality holds when the linear part is a Schottky group.

Modular subgroup arithmetic in free products

Abstract: We study the behaviour modulo p of the number sn(G) of index n subgroups in a group G, where p is an arbitrary prime, and G ranges over a substantial class Cp of free products depending on p. Among other things, we give an explicit combinatorial description of the mod p behaviour of sn(G) for G belonging to a large subclass of Cp, and we investigate the question under which conditions the set {n ∈ N : sn(G) 6≡ 0 (p)}, which captures much of the information contained in the mod p behaviour of sn(G), allows for a characterization in terms of closed formulae. Rather surprisingly, it turns out that Fermat primes play an important special role in the latter question, and, as a byproduct of our investigation, we obtain a new characterization of Fermat primes in terms of the subgroup arithmetic of Hecke groups.

Free subgroups of the group of infinite unitriangular matrices

Abstract: In this note we give simple explicit examples of free subgroups of rank 2 in the group of infinite upper unitriangular matrices over integers. The proofs that the given subgroup is free are elementary. Using canonical projections mod p (p — any odd prime) we obtain also free subgroups in the group of finite state automata on alphabet of size p, extending results of Aleshin [1], Brunner–Sidki [2] and Olijnyk–Sushchansky [8, 9] in this direction. As application, we give the simple proofs of two classical results on free groups.

Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

Abstract: The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.

Algebraic and combinatorial codimension 1 transversality

Abstract: The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen split- tings of CW complexes with fundamental group an injective generalized free product. AMS Classification 57R67; 19J25

Cyclic Statistics In Three Dimensions

Abstract: While 2-dimensional quantum systems are known to exhibit non-permutation, braid group statistics, it is widely expected that quantum statistics in 3-dimensions is solely determined by representations of the permutation group. This expectation is false for certain 3-dimensional systems, as was shown by the authors of ref. [1,2,3]. In this work we demonstrate the existence of ``cyclic'', or $Z_n$, {\it non-permutation group} statistics for a system of n > 2 identical, unknotted rings embedded in $R^3$. We make crucial use of a theorem due to Goldsmith in conjunction with the so called Fuchs-Rabinovitch relations for the automorphisms of the free product group on n elements.

Group representation of the Cayley forest and some of its applications

Abstract: Cayley forests and products of Cayley trees of order are represented as subgroups in the free product of cyclic groups () of order 2. The automorphism groups of these objects are determined. We give a complete description of the sets of translation-invariant and periodic Gibbs measures for the Ising model on a Cayley forest. We construct a new class of limiting Gibbs measures for the inhomogeneous Ising model on a Cayley tree. We find sufficient conditions for random walks in periodic random media on the forest to be never returning provided that the jumps of the walking particle are bounded.

Free products in linear groups

Abstract: Let R be a commutative integral domain of characteristic 0, and let G be a finite subgroup of PGL n (R), the projective general linear group of degree n over R. In this note, we show that if n > 2, then PGL n (R) also contains the free product G * T, where T is the infinite cyclic group generated by the image of a suitable transvection.

Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

Abstract: Let $\ell$ be a length function on a group G, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ 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 $\ell$ 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.

Free products and continuous bundles of C*-algebras

Abstract: The purpose of this thesis is to investigate the properties of free products of C*-algebras and continuous bundles of C*-algebras We also consider how these two areas are connected In the first chapter we present background material relevant to the thesis We discuss nuclearity, exactness and Hilbert C*-modules Then we review the definitions and properties of bundles and free products of C*-algebras The second chapter considers reduced amalgamated free products of C*-algebras We show that, if the initial conditional expectations involved are all faithful, then the resulting free product conditional expectation is also faithful In the third chapter we are interested in the properties of reduced free product C*-algebras We introduce the orthounitary basis concept for unital C*-algebras with faithful traces and show that reduced free products of C*-algebras with orthounitary bases are, except in a few special cases, not nuclear Building on this, we then determine the ideals in a certain tensor product C v Cop of the reduced free product with its opposite C*-algebra In the second half of the chapter, we use Cuntz-Pimsner C*-algebras to study reduced free products of nuclear C*-algebras with respect to pure states We show that, if the GNS representations of the C*-algebras involved contain the compact operators, then the reduced free product C* -algebra is also nuclear Chapter four looks at the minimal tensor product operation on continuous bundles of C*-algebras We construct, for any non-exact C*-algebra C, a continuous bundle A on the unit interval [0,1] such that A C is not continuous This leads to a new characterisation of exactness for C*-algebras These results are then extended to allow for any compact infinite metric space as the base space Finally, we introduce free product operations on bundles of C*-algebras in chapter five Both full and reduced free product bundles are constructed We show that taking the free product (full or reduced) of two continuous bundles gives another continuous bundle, at least when the bundle C*-algebras are exact

A proof of Higgins' conjecture

Abstract: Let f: G=* G(i) -> B=* B(i) be a group homomorphism between free products of groups. Suppose that G(i)f=B(i) of all i. Let H be a subgroup of G such that Hf=B. Then H decomposes into a free product H=*H(i) with H(i)f=B(i). Furthermore, H(i) decomposes into a free product of a free group and the intersection of H(i) with some conjugate of G(i). Higgins conjectured this in 1971 and now we prove it.

On automorphism groups of free products of finite groups, I: Proper Actions

Abstract: If $G$ is a free product of finite groups, let $\Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Gutierrez-Krstic [M. Gutierrez and S. Krstic, {\em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krstic [W. Bogley and S. Krstic, {\em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $\underline{E} \Sigma Aut_1(G)$-space for these groups.

Free Unit Groups in Algebras

Abstract: Let R be an algebra over a field K, and let G be a finite group of units in R. Suppose that either char K = 0 and G is nonabelian, or K is a nonabsolute field of characteristic π > 0 and G/𝕆π(G) is nonabelian. Then we show that there are two cyclic subgroups X and Y of G of prime power order, and two special units u X ∈ KX ⊆ R and u Y ∈ KY ⊆ R, such that ⟨u X , u Y ⟩ contains a nonabelian free group. Indeed, we obtain a rather precise description of these units, generalizing an earlier result where R = K[G] was the group algebra of G over K.

Automorphisms of free product-type and their crossed-products

Abstract: A continuous family of non-outer conjugate aperiodic automor- phisms whose crossed-products are all isomorphic is given on every interpo- lated free group factor. An explicit "duality" relationship between compact co-commutative Kac algebra (minimal) free product actions and free shift actions is also discussed.

On root-class residuality of generalized free products

Abstract: Root-class residuality of free product of root-class residual groups is demonstrated. A sufficient condition for root-class residuality of generalized free product $G$ of groups $A$ and $B$ amalgamating subgroups $H$ and $K$ through the isomorphism $\phi$ is derived. For the particular case when $A=B$, $H=K$ and $\phi$ is the identity mapping, it is shown that group $G$ is root-class residual if and only if $A$ is root-class residual and subgroup $H$ of $A$ is root-class closed. These results are extended to generalized free product of infinitely many groups amalgamating a common subgroup.