Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.     

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

Abelian groups with layered tiles and the sumset phenomenon

We prove a generalization of the main theorem in Jin, The sumset phenomenon, about the sumset phenomenon in the setting of an abelian group with layered tiles of cell measures. Then we give some applications of the theorem for multi-dimensional cases of the sumset phenomenon. Several examples are given in order to show that the applications obtained are not vacuous and cannot be improved in various directions. We also give a new proof of Shnirel'man's theorem to illustrate a different approach (which uses the sumset phenomenon) to some combinatorial problems.

     

Density and invariant means in left amenable semigroups

A left cancellative and left amenable semigroup S satisfies the Strong Følner Condition. That is, given any finite subset H of S and any >0, there is a finite nonempty subset F of S such that for each sH, |sFF|<|F|. This condition is useful in defining a very well behaved notion of density, which we call Følner density, via the notion of a left Følner net, that is a net F_ααD of finite nonempty subsets of S such that for each sS, (|sF_αF_α|)/|F_α| converges to 0. Motivated by a desire to show that this density behaves as it should on cartesian products, we were led to consider the set LIM₀(S) which is the set of left invariant means which are weak^* limits in l_∞(S)^* of left Følner nets. We show that the set of all left invariant means is the weak^* closure of the convex hull of LIM₀(S). (If S is a left amenable group, this is a relatively old result of C. Chou.) We obtain our desired density result as a corollary. We also show that the set of left invariant means on is actually equal to . We also derive some properties of the extreme points of the set of left invariant means on S, regarded as measures on βS, and investigate the algebraic implications of the assumption that there is a left invariant mean on S which is non-zero on some singleton subset of βS.     

On Bredon homology of elementary amenable groups

We show that for elementary amenable groups the Hirsch length is equal to the Bredon homological dimension. This also implies that countable elementary amenable groups admit a finite-dimensional model for of dimension less than or equal to the Hirsch length plus one. Some remarks on groups of type are also made.

     

Generalized notions of amenability, II

This paper continues the investigation of the first two authors begun in part I. It is shown that approximate amenability and approximate contractibility are the same properties, as are uniform approximate amenability and amenability. Bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal. Results are given on Banach sequence algebras, Lipschitz algebras and Beurling algebras, and on the crucial role of approximate identities. A new proof is given for a result due to N. Grønbæk on characterizing of amenability for Beurling algebras.     

On -algebras associated with locally compact groups

Let be a locally compact group, and let denote the same group with the discrete topology. There are various associated to and We are concerned with the question of when these are isomorphic. This is intimately related to amenability. The results can be reformulated in terms of Fourier and Fourier-Stieltjes algebras and of weak containment properties of unitary representations.

     

Ihara's zeta function for periodic graphs and its approximation in the amenable case

In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [B. Clair, S. Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001) 591-620] on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in [R.I. Grigorchuk, A. ?uk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: V.A. Kaimanovich, et al. (Eds.), Proc. Workshop, Random Walks and Geometry, Vienna, 2001, de Gruyter, Berlin, 2004, pp. 141-180] by Grigorchuk and ?uk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

A property of Dunford-Pettis type in topological groups

The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.

In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.

For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

     

New difference sets in nonabelian groups of order 100

In two groups of order 100 new difference sets are constructed. The existence of a difference set in one of them has not been known. The correspondence between a (100, 45, 20) symmetric design having regular automorphism group and a difference set with the same parameters has been used for the construction. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 424–434, 2001     

Self-similar groups in the sense of an iterated function system and their properties

The notion of self-similarity in the sense of iterated function system (IFS) for compact topological groups is given by ?. Koçak in Definition 3. In this work, first we give the definition of strong self-similar group in the sense of IFS. Then, we investigate the main properties of these groups. We also obtain the relations between profinite groups and strong self-similar groups in the sense of IFS. Finally, we construct some examples of these groups.     

Quasi-convex density and determining subgroups of compact abelian groups

For an abelian topological group G, let denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X)<w(G), and an open neighborhood U of 0 in T, we show that . (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map defined by r(χ)=χ?D for , is an isomorphism between and . We prove that

     

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D₂₀ × Z₂ (for example, if G ≃ D₂₀ × K). (2) G has a normal 5‐Sylow subgroup and an elementary abelian 2‐Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5, or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2‐Sylow subgroup S and 5‐Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |ϕ(S)| = 8, and one of the following is true: (a) G = S × T and S is nonabelian; (b) G has a D₁₀ image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in A Γ L(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations of the elementary abelian quotient of order 32 give a contradiction. In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillon's “dihedral trick.” Theorem B summarizes the open nonabelian cases based on this work. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 221–231, 2000     

Algebraic sets and coordinate groups for a free nilpotent group of nilpotency class 2

We give a complete classification of the algebraic sets and coordinate groups for the systems of equations in one variable over a free nilpotent group.