Comparison Theorems and Orbit Counting in Hyperbolic Geometry

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. Introduction In [5] Cannon showed that for fundamental groups of many manifolds of negative curvature (including, for example, compact manifolds) the generating function for word length is a rational function. To explain the implications of this property, we recall that the growth of the quantity N(n) = Card{g ∈ Γ : |g| = n} was studied by Milnor in his fundamental paper [13], where he obtained estimates using comparison with the growth of volume in the universal cover. Cannon used purely combinatorial methods to show that the generating function is rational. In particular, this implies that we can find constants βi and Ci, and positive integers ki (i = 1, . . . , N) such that N(n) = ∑N i=1 Cin iβ i The key step in Cannon’s approach was to associate to the group an automaton. In this article we shall augment this with the notion of “weighting”. This allows us to draw upon the well-known theory of Thermodynamic Formalism to prove a number of new results which can be viewed as weighted analogues of the above results. It has long been understood that this approach must have close connections with symbolic dynamics, Markov partitions, subshifts of finite type (cf. [1], [20],[21] and, in particular, the work of Bourdon [4]) and by extension to the whole paraphernalia of the area of dynamical systems collectively called “Thermodynamic Formalism”. In this article, we shall elaborate on this connection as an integral part of our analysis.