No abstract

Abstract. We consider a group Γ of isometries acting on a (not necessarily geodesic) δ-hyperbolic space and possessing a radial limit set of full measure within its limit set. For any continuous α-quasiconformal measure ν supported on the limit set, we produce a stationary measure µ on Γ. Moreover the limit set together with ν forms a µ-boundary and ν is harmonic with respect to the random walk induced by µ. In the case when X is a CAT(−κ) space and Γ acts cocompactly, for instance, we show that µ has finite first moment. This implies that (∂X, ν) is the unique Poisson boundary for µ. In the course of the proofs, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L 1 or L ∞ norm, for the space of uniformly positive lower-semicontinuous functions on a metric measure space. On the hyperbolic plane H2 we can represent any bounded harmonic function h by the formulafor some f ∈ L ∞ (∂H 2 ) where ∂H 2 is the circle at infinity and ν x are the harmonic measures. Representing H 2 by the unit disk in C, the harmonic measure corresponding to the origin, ν 0 , is just the unit Lebesgue measure on S 1 . The others are given by dν x dν 0 (z) = 1 − |x| 2 |z − x| 2 . The measures ν x also arise from ν 0 by image measures under the transitive isometry group: ν x = g * ν 0 for any g ∈ Isom(H 2 ) such that g(0) = x. Since ν x tends to the dirac measure at z ∈ ∂H 2 as x → z, we obtainIn other words, we have a maximum principle so that the map f → h f determines an isometry, with respect to the L ∞ norms, between the Banach spaces H ∞ (H 2 ) of all bounded harmonic functions and L ∞ (∂H 2 , ν 0 ). Another consequence of the maximum principle is that harmonic functions satisfy the averaging condition. If we identify H 2 with any transitive Lie subgroup of Isom(H 2 ), then at each point x ∈ H 2 any harmonic function h f satisfieswhere µ is the uniform measure on the unit distance circle S ⊂ H 2 around the identity element with respect to the hyperbolic metric. † The author was supported in part by an NSF postdoctoral fellowship and DMS-0306594.‡ The author was supported in part by an NSF postdoctoral fellowship. C. CONNELL AND R. MUCHNIKWe can generalize these concepts greatly to any measure space X with a Markov operator P . We say that a function h is P -harmonic if P h = h. A space B together with a family of mutually absolutely continuous measures {ν x } x∈X is a Poisson Boundary if the map f → h f given by the Poisson formulais an isometry between L ∞ (B, {ν x }) and bounded P -harmonic functions on X. In this event, ν x is called the harmonic measure at x. The Poisson boundary is a purely measurable object and is unique up to measurable isomorphism. As such, even when X is a manifold, the underlying space B could be quite different from that of the topological Martin boundary, which gives a representation space for µ-harmonic functions. The Poisson boundary always exists and has many equivalent descriptions. For instance, it can be identified with the space of ergodic co...

We present an analytic technique for estimating the growth for groups of intermediate growth. We apply our technique to Grigorchuk groups, which are the only known examples of such groups. Our estimates generalize and improve various bounds by Grigorchuk, Bartholdi and others.

Let p c (G) be the critical probability of the site percolation on the Cayley graph of group G. In [2] of Benjamini and Schramm conjectured thatp c < 1, given the group is infinite and not a finite extension of Z. The conjecture was proved earlier for groups of polynomial and exponential growth and remains open for groups of intermediate growth.In this note we prove the conjecture for a special class of Grigorchuk groups, which is a special class of groups of intermediate growth. The proof is based on an algebraic construction. No previous knowledge of percolation is assumed.