Joel M. Cohen

Cogrowth and amenability of discrete groups

Abstract Let G be a group and g1,…, gt a set of generators. There are approximately (2t − 1)n reduced words in g1,…, gt, of length ⩽n. Let \ gg n be the number of those which represent 1G. We show that γ = lim n → ∞ ( \ gg n ) 1 n exists. Clearly 1 ⩽ γ ⩽ 2t − 1. η = ( log γ) ( log (2t − 1)) is the cogrowth. 0 ⩽ η ⩽ 1. In fact η ∈ {0} ∪ ( 1 2 , 1¦ . The entropic dimension of G is shown to be 1 − η. It is then proved that d(G) = 1 if and only if G is free on g1,…, gt and d(G) = 0 if and only if G is amenable.

Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis

On calcule la mesure pour laquelle une suite de polynomes avec une formule de recurrence constante est orthogonale

The 2-circle and 2-disk problems on trees

Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σnf(x) be the sum Σd(x, y)=nf(y) and letBnf(x)=Σd(x, y)≦nf(y). The purpose is to study to what extent Σnf andBnf determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σnf orBnf vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σn andBn respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.

C∗-algebras without idempotents

The simplest statement of the main results are these: Let π be a free group on 2 generators. Let Cπ be the complex ring and L1π the ring extension to L1 sums. Then L1π contains no idempotents. Furthermore, if α ϵ Cπ, β ϵ L1π are nonzero, then αβ ≠ 0. The first result is in the direction of proving that a certain simple C∗-algebra has no idempotents yielding a counter-example to the suggestion that simple C∗-algebras are generated by their projections.

Embeddings of trees in the hyperbolic disk

We consider the conditions under which a homogeneous tree of even degree (i.e. the Cayley graph of a free group) may be embedded in the hyperbolic disk in such a way that the automorphisms induced by rotation and by translation by group elements may be represented by automorphisms of the disk. We concentrate on the study of a particular optimal such embedding. We show that in this case bounded analytic functions on the disk are determined by their values at the vertices of the embedded tree. Using a construction of we can define harmonicity on a homogeneous tree in such a way that bounded functions on the disk restrict to harmonic functions on the tree if and only if they are harmonic. We also show a simple and elegant construction of this tree.

POLYHARMONIC FUNCTIONS ON TREES

In this paper, we introduce and study polyharmonic functions on trees. We prove that the discrete version of the classical Riquier problem can be solved on trees. Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees. Furthermore, on a homogeneous tree we characterize the polyharmonic functions in terms of integrals with respect to finitely-additive measures (distributions) of certain functions depending on the Poisson kernel. We define polymartingales on a homogeneous tree and show that the discrete version of a characterization of polyharmonic functions due to Almansi holds for polymartingales. We then show that on homogeneous trees there are 1-1 correspondences among the space of nth-order polyharmonic functions, the space of nth-order polymartingales, and the space of n-tuples of distributions. Finally, we study the boundary behavior of polyharmonic functions on homogeneous trees whose associated distributions satisfy various growth conditions.

Function Series, Catalan Numbers, and Random Walks on Trees

The delight of finding unexpected connections is one of the rewards of studying mathematics. In this talk, based on joint work with Ibtesam Bajunaid, Joel Cohen, and David Singman, I will discuss the connections that link the following seven superficially unrelated entities: (A) A function of the sort that calculus textbooks often use to show that a continuous function need not have a derivative at each point:

Bounded holomorphic functions on bounded symmetric domains

Let D be a bounded homogeneous domain in C , and let A de- note the open unit disk. If z e D and /: D —► A is holomorphic, then s/(z) is defined as the maximum ratio \Vz(f)x\/Hz(x, 3c)1/2 , where x is a nonzero vector in C and Hz is the Bergman metric on D. The number sf(z) rep- resents the maximum dilation of / at z . The set consisting of all s/(z), for z e D and /: D —► A holomorphic, is known to be bounded. We let cr, be its least upper bound. In this work we calculate Cr, for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions / for which sf = C£>) when D contains A as a fac- tor, and show that the class of extremal functions is very large when A is not a factor of D. Definition 1. Let M and N be Riemannian manifolds and /: M -* N a smooth function. Then / induces a map of the tangent bundles, df: TM —► TN. At each point x £ M, dfx: TXM —► Tf^X)N is a linear transformation of Euclidean spaces. Define s/(x) as the norm of dfx considered as map of normed vector spaces. We call / Bloch if the constant sf = supx€M s/(x) is finite. The number sf is called the Bloch constant of /. It is easy to see that a smooth Lipschitz function (i.e., a function with bounded dilation) is necessarily Bloch. In fact given x £ M, if X is a pos- itive constant such that d^(f(x), f(y)) < Xd?f(x, y), for every y £ M, then sf(x) < X. Here dM and dN denote the distance functions induced by the Riemannian metrics of M and N, respectively.

Radial functions on free products

On considere les groupes dont les fonctions radiales forment une algebre. On decrit les mesures de Plancherel sur une grande classe de ces groupes radiaux

A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbor transition probability that yields a recurrent random walk and show that, although such trees have no positive potentials, many of the standard results of potential theory can be transferred to this setting. We accomplish this by defining a non-negative function H, harmonic outside the root e and vanishing only at e, and a substitute notion of potential which we call H-potential. We define the flux of a superharmonic function outside a finite set of vertices, give some simple formulas for calculating the flux and derive a global Riesz decomposition theorem for superharmonic functions with a harmonic minorant outside a finite set. We discuss the connection of the H-potentials with other notions of potentials for recurrent Markov chains in the literature