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
Ð´Ð³Ð¾Ð¸Ñtheory. 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
ÑÐ°ÑÐµpoint. His work was confined to the use of generators with geometric
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.