Fixed Points in Boundaries of Negatively Curved Groups

An important feature of the natural action of a negatively curved (= word hyperbolic) group on its boundary is that the xed points of hyperbolic (= innnite order) elements form a dense subset of the boundary. Gromov states this in his innuential treatise Hyperbolic Groups 4, 8.2 D] and his explanation involves Stalling's theorem on groups with innnitely many ends. This short note provides an elementary proof of this fact. The proof relies on a simple enhancement of the Pumping Lemma from the theory of nite state automata and on the fact that the geodesic words with respect to a xed generating set for a negatively curved group form a regular language. For background material on negatively curved groups see 1] and on automata and regular languages on groups see 2]. For a diierent proof see 5]. Deenition. Let L be a language over the nite alphabet A. An innnite sequence fa i g of letters from A is an L-ray provided, for every integer n > 0, there exists an integer N n such that a 1 a N is a word in the language L.