Symbolic dynamics and hyperbolic groups. (English) Zbl 0783.58017

Coornaert, Michel; Papadopoulos, Athanase

Lecture Notes in Mathematics. 1539. Berlin: Springer-Verlag. viii, 138 p. (1993).

The book explains aspects of Gromov’s theory of hyperbolic spaces and hyperbolic groups from the point of view of symbolic dynamics. Symbolic dynamics is a way of topologically describing a dynamical system by semi- conjugating it to a subshift of finite type. Depending on the properties of this semi-conjugacy one can classify the dynamical system as being of finite type, or finitely presented, etc. In any case a combinatorial model of the dynamics is provided. The dynamical system studied in the book is a hyperbolic group acting on its own hyperbolic boundary. Two direct constructions of symbolic dynamics are provided which make this a finitely presented dynamical system. This description is later used to obtain a combinatorial model of the hyperbolic boundary (of a torsion- free hyperbolic group) as a so-called semi-Markovian space. The book provides a careful and complete account of a part of a vigorously developing subject. A summary of Gromov’s theory of hyperbolic spaces is provided, as well as a summary of facts about symbolic dynamics. This way, the book should be accessible to a geometer eager to learn how the formalism of symbolic dynamics works, as well as to a dynamicist interested in hyperbolic groups.

Reviewer: G.Swiatek (Stony Brook)

MSC:

37E99	Low-dimensional dynamical systems
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10	Fuchsian groups and their generalizations (group-theoretic aspects)
58-02	Research exposition (monographs, survey articles) pertaining to global analysis

Keywords:

finitely presented system; hyperbolic spaces; hyperbolic groups; symbolic dynamics; subshift of finite type; Gromov’s theory