The Poisson formula for groups with hyperbolic properties. (English) Zbl 0984.60088

Kaimanovich, Vadim A.

Ann. Math. (2) 152, No. 3, 659-692 (2000).

Given a Markov operator \(P\) on a Lebesgue measure space \((X,m)\), there exists a space \(\Gamma\) (called the Poisson boundary of \(P\)) equipped with a family of probability measures \(\nu_x\), \(x\in X\), such that the Poisson formula establishes an isometry between the space of \(P\)-harmonic functions (that is, functions \(f\) satisfying \(Pf=f\)) from the space \(L^{\infty}(X,m)\) to the space \(L^{\infty}(\Gamma)\). The Poisson boundary is defined as the space of ergodic components of the time shift \(T\) in the space of sample paths of the Markov chain of \(X\) associated with the operator \(P\), the measures \(\nu_x\) being the images of the measures in the path space corresponding to starting the Markov chain at points \(x\in X\). In the case where the space \(X\) is endowed with additional (geometric, algebraic, etc.) structures, the operator \(P\) is supposed to have some properties which make it compatible with this structure, and it is natural to ask for a description of the Poisson boundary (which in principle is a measure-theoretic object) in terms of this structure. This paper addresses the question of identifying the Poisson boundary for the random walk operator determined by a probability measure \(\mu\) on a countable group \(G\). The author develops new methods and obtains new results on the subject, generalizing previous results of himself and of other authors. The methods consist in describing the Poisson boundary of certain groups, via entropy estimates for conditional random walks. The boundaries are identified then as natural topological boundaries. The classes of groups concerned are those of word hyperbolic groups and discontinuous groups of isometries on Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan-Hadamard manifolds and discrete subgroups of semi-simple Lie groups.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

60J50	Boundary theory for Markov processes
28D20	Entropy and other invariants
53C22	Geodesics in global differential geometry
31C05	Harmonic, subharmonic, superharmonic functions on other spaces
20F67	Hyperbolic groups and nonpositively curved groups

Keywords:

Poisson boundary; Markov chain; Markov operator; path space; cocompact lattice; hyperbolic group; harmonic function; entropy; conditional process; end of a group; Cartan-Hadamard manifold; semi-simple Lie group; discrete subgroup; Furstenberg boundary