Translation numbers in negatively curved groups (Q1971129)

Let \(G\) be a finitely presented group with word metric \(d\). The translation number of an element \(g\) of \(G\) is \[ \tau(g)=\lim_{n\to\infty}{|g^n|\over n}, \] where \(|g|=d(1,g)\). The author presents a short proof of the claim of Gromov that the translation numbers of elements in a word hyperbolic group are rational with bounded denominator. The proof provides a bound on the denominator for all elements in terms of the number of states in the finite state automaton accepting the language of geodesic words.