Abstract. We show that the set SA(G) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group G is dense in the product of the sets SA(P) over all maximal parabolic subgroups P. The set BSA(G) of equivalence classes of biautomatic structures on G is isomorphic to the product of the sets BSA(P) over the cusps (conjugacy classes of maximal parabolic subgroups) of G. Each maximal parabolic P is a virtually abelian group, so SA(P) and BSA(P) were computed in [NS1]. We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics for G is regular. Moreover, the growth function of G with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are. 1.

We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoulli-p and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain fi of small degree, simulation gives the Hausdorff dimension to several decimal places.

. Automatic groups were introduced in connection with geometric problems, in particular with the study of fundamental groups of 3-manifolds. In this article the class of automatic groups is extended to include the fundamental group of every compact 3-manifold which satisfies Thurston's geometrization conjecture. Toward this end the class C A of asynchronously A-combable groups is introduced and studied, where A is an arbitrary full abstract family of languages. For example A may be the family of regular languages Reg, context--free languages CF, or indexed languages Ind.TheclassC Reg consists of precisely those groups which are asynchronously automatic. It is proved that C Ind contains all of the above fundamental groups, but that C CF does not. Indeed a virtually nilpotent group belongs to C CF if and only if it is virtually abelian. Introduction In the last several years there has been a remarkable interplay between topology, geometry, group theory, and the t...

It is known that the Cayley graph \Gamma of a negatively curved (Gromov-hyperbolic) group G has a well-defined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 , then G is topologically conjugate to a cocompact Fuchsian group.

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk

In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map [.] : G!/G=[G;G]. This is obtained via an analysis involving subshifts of nite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus >= 2 is also discussed.

3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65

In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both "thermodynamic" ergodic theory and the automaton associated to strongly Markov groups. 0.

Abstract. Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [12]. Using the definition of Farb of a relatively hyperbolic group [9], we prove this assertion. We conclude that the conjugacy problem is solvable for the following two classes of groups: fundamental groups of complete, finite-volume, negatively curved manifolds, and finitely generated fully residually free groups.

Abstract. We prove that the language of all geodesics of any Garside group, with respect to the generating set of divisors of the Garside element, forms a regular language. In particular, the braid groups admit generating sets where the associated language of geodesics is regular. 1.