Ergodic Theory and Dynamical Systems



Groups of automorphisms of trees and their limit sets


SA'AR HERSONSKY a1 and JOHN HUBBARD a1
a1 Department of Mathematics, Cornell University, Ithaca NY 14853-7901, USA (e-mail: saar@math.cornell.edu and hubbard@math.cornell.edu)

Abstract

Let $T$ be a locally finite simplicial tree and let $\Gamma\subset{\rm Aut}(T)$ be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincaré series associated with $\Gamma$, which is also the Hausdorff dimension of the limit set of $\Gamma$; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of $\Gamma$. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its dimension; we give a new proof of this result. Finally, we show that the critical exponent is locally constant on the space of deformations of $\Gamma$.

(Received May 15 1995)
(Revised October 17 1995)