Symbolic Dynamics and Hyperbolic Groups

Michel Coornaert, Athanase Papadopoulos (auth.)3540564993, 9783540564997, 0387564993

Gromov’s theory of hyperbolic groups have had a big impact in combinatorial group theory and has deep connections with many branches of mathematics suchdifferential geometry, representation theory, ergodic theory and dynamical systems. This book is an elaboration on some ideas of Gromov on hyperbolic spaces and hyperbolic groups in relation with symbolic dynamics. Particular attention is paid to the dynamical system defined by the action of a hyperbolic group on its boundary. The boundary is most oftenchaotic both as a topological space and as a dynamical system, and a description of this boundary and the action is given in terms of subshifts of finite type. The book is self-contained and includes two introductory chapters, one on Gromov’s hyperbolic geometry and the other one on symbolic dynamics. It is intended for students and researchers in geometry and in dynamical systems, and can be used asthe basis for a graduate course on these subjects.

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Table of contents :
Introduction….Pages 1-4
A quick review of Gromov hyperbolic spaces….Pages 5-18
Symbolic dynamics….Pages 19-42
The boundary of a hyperbolic group as a finitely presented dynamical system….Pages 43-68
Another finite presentation for the action of a hyperbolic group on its boundary….Pages 69-90
Trees and hyperbolic boundary….Pages 91-106
Semi-Markovian spaces….Pages 107-117
The boundary of a torsion-free hyperbolic group as a semi-Markovian space….Pages 118-134