Donald I. Cartwright

Groups acting simply transitively on the vertices of a building of typeÃ2, I

If a group Γ acts simply transitively on the vertices of an affine building Δ with connected diagram, then Δ must be of typeÃn−1 for somen⩾2, and Γ must have a presentation of a simple type. The casen=2, when Δ is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when Δ is the building ofG=PGL(3,K),K a local field, and when Γ⩽G. We exhibit such a group Γ whenK=Fq((X)),q any prime power. Our study leads to combinatorial objects which we calltriangle presentations. These triangle presentations give rise to some new buildings of typeÃ2.

Random walks on free products, quotients and amalgams

Suppose that G is a discrete group and p is a probability measure on G . Consider the associated random walk {X n } on G . That is, let X n = Y 1 Y 2 … Y n , where the Y j ’s are independent and identically distributed G -valued variables with density p . An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p . For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x ) permits a description of the asymptotic behaviour of these probabilities.

Harmonic analysis for groups acting on triangle buildings

Let Δ be a thick building of type A 2 , and let be its set of vertices. We study a commutative algebra of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi- K -invariant functions on G , if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C *-algebra induced by on l 2 , find its spectrum Σ, prove positive definiteness of a kernel k z for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings Δ J arising from the groups Γ J introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of Γ J .

Martin and end compactifications for non-locally finite graphs

We consider a connected graph, having countably infinite vertex set X, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix P corresponding to a nearest neighbor random walk on X, we study the associated harmonic functions on X and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of X, the set of ends, and the set of improper vertices-new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many- generators

A family of à n -groups

AbstractFor each integern≥2 and for each prime powerq, we exhibit a group Γ which acts simply transitively on the set of vertices of the building of type Ãn associated with the local field

Harmonic analysis on the free product of two cyclic groups

\mathbb{F}^q ((Y))

Infinite graphs with nonconstant Dirichlet finite harmonic functions

. This generalizes work in [2] and [8] (wheren=2) and in [1] (wheren≤4).

Ramanujan geometries of type &Ã n

Let μ be a probability on a free group Γ of rank r ≥2. Assume that Supp(μ) is not contained in a cyclic subgroup of Γ. We show that if (X n )n≥0 is the right random walk on Γ determined by μ, then with probability 1, X n converges (in the natural sense) to an infinite reduced word. The space Ω of infinite reduced words carries a unique probability ν such that (Ω, ν) is a frontier of (Γ, μ) in the sense of Furstenberg. This result extends to the right random walk (X n ) determined by a probability μ on the group G of automorphisms of an arbitrary infinite locally finite tree T