Poincaré Series and Zeta Functions for Surface Group Actions on R-trees

and it is easy to see that this is constant on conjugacy classes. In this setting, it can be shown that η(s) and ζ(s) have extensions as meromorphic functions to the entire complex plane. The proof relies on non-commutative harmonic analysis and the functions are studied via the spectral properties of the Laplace-Beltrami operator. In this note we shall consider an analogous situation where we replace H by an R-tree. R-trees are a class of metric spaces which generalize the more familiar simplicial trees. In recent years there has been much interesting work on group actions on R-trees (for a good survey see [10]) which can, in part, be viewed as a generalization of the now classical Bass-Serre theory of group actions on trees [16]. In particular, Morgan and Shalen have shown that the fundamental groups Γ of compact surfaces M with Euler characteristic strictly less than −1 act freely on R-trees. More precisely, they show that given a hyperbolic structure on M , there