Connections between the theory of hyperbolic manifolds and the theory of automata are deeply interwoven in the history of mathematics of this century.

The use of symbol sequences to study dynamical systems originates in the work of Kocbe [Koc27, Koe29] and Morse [Mor87j, who both used symbol saliences to code geodesies on a surface of constant, negative curvature. Er godic theorists have been motivated by the consideration of geodesic flows on hyperbolic manifolds, amongst, other things, to consider shifts of finite type. The concept of a "sophic system" is simply the ergodic theorist's specialized name for what is known in other branches of science as a finite state automa ton. In [BS79], Bowen and Series describe Markov partitions related to the action of certain groups of hyperbolic isometries on the boundary space.

Another fundamental contribution was made by Max Dehn [Deh87]. He was the first person to point out the importance of the word problem in дгоирtheory. His solution in the case of fundamental group of a surface is very much in the spirit of what we are doing, namely a geometric approach lo group theory. He was aware that geodesies in the Cayley graph of such a group follow certain rules (formalized in this book by the use of a finite state automaton), and that rules also characterize pairs of geodesies ending at the чашеpoint. His work was confined to the use of generators with geometric significance.

In 1978 Thurston and Cannon had a conversation at the International Congress of Mathematicians in Helsinki about growth functions of groups, in which Thurston conjectured that, the growth function of a hyperbolic group is rational. Cannon had already worked out. cone types (page 66) for a number of examples of group presentations, and he later discovered that his coinputa »i<mis of cone types could easily be used to find growth functions and show that they were rational. Further investigations by Cannon [Can84] showed once again the connections between recursive processes and hyperbolic phenom ena. Thurston observed that Cannon's results had an appropriate expression in terms of finite state automata in two variables.