Stable Exponents in Discrete Groups

It is known that in a word hyperbolic group the stable exponent of every nontorsion element is an integer. We prove that this is also true in finitely generated nilpotent groups. On the other hand, we show that for any rational number 1 there exists a torsionfree CAT(0) group containing an element whose stable exponent is equal to .     

Spherical functions and conformal densities on spherically symmetric -spaces

Let be a -space which is spherically symmetric around some point and whose boundary has finite positive dimensional Hausdorff measure. Let be a conformal density of dimension on . We prove that is a weak limit of measures supported on spheres centered at . These measures are expressed in terms of the total mass function of and of the dimensional spherical function on . In particular, this result proves that is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.

     

Injective linear cellular automata and sofic groups

Let V be a finite-dimensional vector space over a field and let G be a sofic group. We show that every injective linear cellular automaton τ: V ^G → V ^G is surjective. As an application, we obtain a new proof of the stable finiteness of group rings of sofic groups, a result previously established by G. Elek and A. Szabó using different methods.     

Positive Eigenfunctions of the Laplacian and Conformal Densities on Homogeneous Trees

In this paper, we study some asymptotic aspects of the positiveeigenfunctions of the combinatorial Laplacian associated toa homogeneous tree. The results are inspired by results of DennisSullivan concerning -harmonic functions on the hyperbolic spacesHⁿ and contained in the paper [9].     

Amenability and Linear Cellular Automata Over Semisimple Modules of Finite Length

Let M be a semisimple left module of finite length over a ring R and let G be an amenable group. We show that an R-linear cellular automaton τ:M^G → M^G is surjective if and only if it is pre-injective.     

An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups

We prove an analogue of Fekete’s lemma for subadditive right-subinvariant functions defined on the finite subsets of a cancellative left-amenable semigroup. This extends results previously obtained in the case of amenable groups by E. Lindenstrauss and B. Weiss and by M. Gromov.     

A note on stable exponents

Given an element γ in a group γ, the stable exponent p+(γ) of γ is defined as p+(γ) =lim sup_n→∞P(γⁿ) denotes the exponent of P(γⁿ) = sup{k/ ?γ_o ∈ γ such that γⁿ = γ^k _o We prove that if γ acts properly discontinuously by isometrics on a proper geodesic Gromov-hyperbolic metric space and γ ∈ γ is of hyperbolic type, then P+(γ) is an integer. This implies that the stable exponent of every element of infinite order in a word hyperbolic group is an integer. We also show that, in a translation discrete group, the stable exponent of every element of infinite order is finite.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.