The ergodic theory of discrete groups. (English) Zbl 0674.58001

Nicholls, Peter J.

London Mathematical Society Lecture Note Series, 143. Cambridge etc.: Cambridge University Press. xi, 221 p. £19.50; $ 34.50 (1989).

This book gives an introduction to recent developments in the theory of discrete groups \(\Gamma\) acting on a hyperbolic space \({\mathbb{H}}\). The class of groups under consideration is that of geometrically finite ones, and therefore includes many groups of infinite covolume. In these cases the associated geodesic flow on the unit sphere bundle of \(\Gamma\) \(\setminus {\mathbb{H}}\) is dissipative but nevertheless the dynamics of the non-wandering set are very susceptible to the techniques of modern ergodic theory. The Gibbs measure for the appropriate (logarithm of a) power of the conformal dilation is given in terms of a “conformal density”, a measure with good transformation properties on the limit set. It turns out that this measure is often, but not always, a Hausdorff measure and is therefore one of the few examples of a Hausdorff measure actually appearing “in nature”. The Gibbs measure is actually ergodic, a fact which generalizes Hopf’s famous “ergodic theorem” and which has many important applications.
The book under review gives a solid introduction to this circle of ideas and covers the foundations carefully.
The subject matter is as follows:
- the basic geometry of discrete groups,
- those aspcts of ergodic theory needed for the study of discrete groups,
- the fundamental concepts associated with the limit set and the “critical exponent”,
- the construction and analysis of a conformal density on the limit set up to the generalization of Hopf’s theorem and the determination of the Hausdorff dimension of the limit set, and
- some questions special to the theory of Fuchsian groups.

Reviewer: S.J.Patterson

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
37A99	Ergodic theory
20H10	Fuchsian groups and their generalizations (group-theoretic aspects)
30F35	Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
37D99	Dynamical systems with hyperbolic behavior
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25	Geodesic flows in symplectic geometry and contact geometry
22E99	Lie groups

Keywords:

discrete groups; hyperbolic space; Gibbs measure; ergodic theory; limit set; critical exponent; conformal density; Hopf’s theorem; Hausdorff dimension; Fuchsian groups