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<ul><li><p>GAFA, Geom. funct. anal.Vol. 17 (2007) 707 7691016-443X/07/030707-63DOI 10.1007/s00039-007-0608-9ONLINE FIRST: May 2007</p><p>c Birkhauser Verlag, Basel 2007</p><p>GAFA Geometric And Functional Analysis</p><p>HARMONICITY OF QUASICONFORMAL MEASURESAND POISSON BOUNDARIES OF HYPERBOLIC</p><p>SPACES</p><p>Chris Connell and Roman Muchnik</p><p>Abstract. We consider a group of isometries acting on a proper(not necessarily geodesic) -hyperbolic space X . For any continuous -quasiconformal measure on X assigning full measure to r, the radiallimit set of , we produce a (nontrivial) measure on for which is sta-tionary. This means that the limit set together with forms a -boundaryand is harmonic with respect to the random walk induced by . As abasic example, take X = Hn and to be any geometrically nite Kleiniangroup with a PattersonSullivan measure for . In the case when Xis a CAT(1) space and is discrete with quasiconvex action, we showthat (r, ) is the Poisson boundary for . In the course of the proofs,we establish sucient conditions for a set of continuous functions to forma positive basis, either in the L1 or L norm, for the space of uniformlypositive lower-semicontinuous functions on a general metric measure space.</p><p>On the hyperbolic plane H2, we can represent any bounded harmonic func-tion h by the formula</p><p>h(x) = h(x) =H2</p><p>dx</p><p>for some L(H2), where H2 is the circle at innity and x is thegeodesic projection to H2 of the Lebesgue measure on the unit tangentcircle SxH2. Representing H2 by the unit disk in C, the harmonic measurecorresponding to the origin, 0, is just the unit Lebesgue measure on S1.The others are given by integrating against the Poisson kernel,</p><p>dxd0</p><p>(z) =1 |x|2|z x|2 .</p><p>Keywords and phrases: Hyperbolic group, Poisson boundary, Gibbs state, randomwalks</p><p>AMS Mathematics Subject Classification: 60J50, 20F67, 37A35, 41A65The first author was supported in part by an NSF postdoctoral fellowship and DMS-</p><p>0420432. The second author was supported in part by an NSF postdoctoral fellowship.</p></li><li><p>708 C. CONNELL AND R. MUCHNIK GAFA</p><p>The measures x also arise as the translates of 0 by the isometry group:x = g0 for any g Isom(H2) such that g(0) = x.</p><p>The measure x tends weakly to the Dirac measure at z H2 as x z.Hence for any continuous : H2 R, we have</p><p>hL(H2) = ess supxH2</p><p>H2</p><p>(z)dx(z) = L(H2) .</p><p>More generally, a classical result states that the mapping h givesan isometry between L(H2, 0) and the space H(H2) of all boundedharmonic functions.</p><p>Another well-known property of harmonic functions is that they satisfyan averaging condition. At each point x H2 any harmonic function hsatises</p><p>h(x) =B(x,r)</p><p>h(y)d(y) ,</p><p>where B(x, r) H2 is the ball of radius r centered at x and is theRiemannian volume measure normalized by (B(x, r)) = 1.</p><p>We can generalize these concepts greatly to any measure space X witha Markov operator P . We say that a measurable function h is P -harmonicif Ph = h. A space B together with a family of mutually absolutely contin-uous measures {x}xX is called the Poisson boundary if the map hgiven by the Poisson formula</p><p>h(x) :=B</p><p>(y)dx(y) (0.1)</p><p>is an isometry between L(B, {x}) and bounded P -harmonic functionson X. In this event, x is called the harmonic measure at x. The Poissonboundary is a purely measurable object and is unique up to measurableisomorphism. The Poisson boundary always exists and has many equivalentdescriptions. For instance, it can be identied with the space of ergodiccomponents of the shift map T acting on the space of sample paths ofthe Markov chain on X associated with the operator P . The harmonicmeasures x are the images under the quotient map of the measures Pxin the path space corresponding to starting the Markov process from statex X. For other characterizations of the Poisson boundary see [K3].</p><p>Before moving to random walks on groups, we recall how to form theconvolution of two measures. Suppose (X, ) is a measure space and isany set of -measurable transformations of X. For any measure on wedene the convolution of the two measures to be the measure on Xgiven by</p><p> := d() ,</p></li><li><p>Vol. 17, 2007 HARMONIC MEASURES 709</p><p>where (A) = (1A). As a special case of the above constructions,we can dene the (right-sided) random walk determined by a probabil-ity measure on a group G as follows. Let GZ+ =</p><p>i=1 G and denote</p><p>by P the measure obtained as the image of under the map(x1, x2, x3, . . . ) (e, x1, x1x2, x1x2x3, . . . ). The conditional measure for Pon the n-th coordinate of GZ+ is the n-fold convolution measure n = n</p><p>on G. The measure on the space of sample paths with starting point givenby the initial distribution is P. In this context, the natural Markov op-erator associated to the random walk is P : L(G,) L(G,) denedby</p><p>P(f)(g) =G</p><p>f(gg)d(g) .</p><p>In this setting we call P-harmonic functions simply -harmonic. More-over, for the Poisson boundary (B, {g}gG) it follows that g = ge sowe may simply write the boundary as (B, ) where = e. It follows fromthe denition of -harmonicity and the Poisson formula (0.1) that for all L(B, ) at e, we have</p><p>() = h(e) =G</p><p>h(g)d(g) =G</p><p>g()d(g) .</p><p>In short, = , in which case we also say is -stationary. Theimportance of -stationary measures is that the Poisson formula (0.1) yields-harmonic functions: for any g G,G</p><p>h(gg)(g) =G</p><p>B</p><p>(z)d(gg)(z)d(g)</p><p>=B(z)dg</p><p>(G</p><p>g d(g))(z) =</p><p>B(z)dg(z) = h(g) .</p><p>A measured G-space (B, ) is called a -boundary of G if it is aG-equivariant quotient of the Poisson boundary. This is equivalent tothe statement that for P-a.e. path = g1g2 . . . in G, the measuresgi weak- converge to a single point measure on B. This implies that = and that the corresponding Poisson formula, h using ,denes an isometric embedding from L(B, ) into the space of bounded-harmonic functions H(G,). Any -boundary arises as a G-equivariantmeasurable quotient : (B, ) (B, ) of the Poisson boundary (B, )since the induced lift map : L(B, ) L(B, ) is an isometricembedding. In particular, the Poisson boundary can be characterized asthe maximal -boundary (see the unpublished survey [K3]). On the otherhand, it does not follow that any G-space with a -stationary measureis isomorphic to a -boundary. However, such a G-space (B, ) will be</p></li><li><p>710 C. CONNELL AND R. MUCHNIK GAFA</p><p>a -boundary if the action of G on (B, ) is -proximal in the sense ofFurstenberg (see [Fu4]). This characterization can be useful for identifying-boundaries as subsets of geometrically dened boundaries for G.</p><p>For example, consider the case G = Isomo(H2) = PSL(2,R) with amaximal compact subgroup K = Stab(o) = PSO(2) for a xed basepointo H2. Let mK denote the (bi-invariant) Haar measure on K which we willthink of as a measure on G supported on K, and choose an element g Gsuch that d(o, go) = 1. If 0 = mKgmK , then bounded 0-harmonic func-tions on G are right K invariant and so their quotients live in G/K = H2. If : G G/K is the quotient map, then by construction 0 is uniformlydistributed on the unit circle about o so that the 0-harmonic functionssatisfy the averaging property. Hence 0-harmonic functions descend to theclassical harmonic functions on H2. Moreover, every 0 harmonic functionis the lift of one on H2. In particular, (H2, 0) is the Poisson boundary of(G,0).</p><p>In fact this same correspondence was established by Furstenberg in [Fu1]for any symmetric space G/K where G is semisimple Lie group of noncom-pact type and K is a maximal compact subgroup. If we again take the bi-Kinvariant measure 0 = mK gmK , then the Poisson boundary of (G,0) isthe Furstenberg boundary G/P , where P is a minimal parabolic subgroup,together with the unique K-invariant measure 0. Later, Furstenberg ex-tended this in [Fu2] and [Fu3] to show that for any lattice < G, one canbuild a measure on for which (G/P, 0) is the Poisson boundary. Geo-metric intuition makes it tempting to believe that passing from a Lie groupto a lattice, at least a uniform lattice, should be an operation garneringidentical asymptotic information of any kind. However, the measure con-structed by Furstenberg is quite dierent from the measure 0 on G. Forinstance, there are no K-invariant measures on the lattice and the measure need not be compactly supported. In the rank one case, we will showthat there is an innite dimensional space of measures on for which(G/P, 0) is the Poisson boundary.</p><p>Furstenberg proved this result for in two steps. First he constructeda for which 0 was -stationary. By showing that acts -proximally,he concluded that this was a -boundary. Next he showed that if hasnite rst moment, then (G/P, 0) is the Poisson boundary. Here niterst moment means, </p><p>()d(e, ) < .</p><p>He used this geometric characterization of the Poisson boundary to</p></li><li><p>Vol. 17, 2007 HARMONIC MEASURES 711</p><p>distinguish envelopes of certain discrete subgroups. Specically, a discretegroup < SL(2,R) cannot be a lattice in SL(n,R) for n > 2.</p><p>Kaimanovich and Vershik in [KV] gave sucient and necessary condi-tions for the Poisson boundary of a random walk on a discrete group tobe trivial. In [K2], Kaimanovich generalized this to give criteria to decidewhen a certain geometric boundary for a group together with a family ofexit measures could be the Poisson boundary for a given random walk onthe group. For example, we may consider the special case of a nonelemen-tary -hyperbolic group , where nonelementary means # = . If carries a -stationary measure , he showed, e.g. Theorem 7.7 of [K2], that(, ) is the Poisson boundary for (, ) provided has both nite rstlog-moment and nite entropy:</p><p>() log d(e, ) < and h() := </p><p>() log () < .</p><p>The goal of the present paper is to generalize Furstenbergs results togroups acting isometrically on a Gromov hyperbolic space and to a generalclass of boundary measures. By so doing, we can partially answer a conversequestion to that answered by Kaimanovich and Vershiks results statedabove. Namely, starting with a measure on can we nd a measure on such that (, ) is the Poisson boundary of ? In fact, not every measure can arise as the Poisson boundary, and there are examples of measures which do arise but f does not for certain positive measurable functions f(see Remark 6.3). Nevertheless, we shall give an armative answer forany measure Lipschitz equivalent to a PattersonSullivan measure on aCAT(1) group. If one asks the same question for -boundaries instead,then we show existence for continuous measures in this class on a largefamily of groups which includes the Gromov hyperbolic groups.</p><p>In a second paper, we will broaden some of these results to multiplemeasure classes within the family of Gibbs streams. While we presentlyrestrict our attention to the hyperbolic setting, we hold out the hope thatin the future we can use our techniques to answer related questions for somenonamenable groups which are almost nonpositively curved in some sense.</p><p>We now state some theorems highlighting the kinds of results we areable to achieve with our methods. The next theorem is a special case ofTheorem 1.19. We point out that the PattersonSullivan measures are themost widely-known examples of -quasiconformal measures (see section 1).</p><p>Theorem 0.1 (Harmonicity). Let X be a proper geodesic -hyperbolicspace and suppose < Isom(X) acts cocompactly on X. Let be a</p></li><li><p>712 C. CONNELL AND R. MUCHNIK GAFA</p><p>continuous -quasiconformal measure on X, for any > 0. If f is a posi-tive lower semi-continuous function on X, then there exists a (nontrivial)measure on such that = f.</p><p>The main signicance of the above theorem is the following corollarywhich follows by completely general and well-known arguments from The-orem 0.1, e.g. see [K2].</p><p>Corollary 0.2. If X, and are as in the previous theorem, then(X, f) is a -boundary of .</p><p>Even in the case when X is a simply connected negatively curved man-ifold, it does not follow that (X, ) is the Poisson boundary for . Inother words, there may exist a nonzero -harmonic function which van-ishes asymptotically -a.e. on X. In fact, it is unknown whether any exists making (X, ) the Poisson boundary for (, ). To guarantee themaximality of the above boundary we need to connect the large scale be-havior of to the large scale behavior of the metric on X. Kaimanovich[K2] has formulated very general criteria for establishing maximality of -boundaries for discrete groups. In the case when X is a CAT() space, weare able to establish these criteria under a few additional assumptions. Thisyields our second principle result which is a special case of Theorem 1.22.</p><p>Theorem 0.3 (Poisson boundary). Suppose that, in addition to thehypotheses of Theorem 0.1, we assume X is a CAT(1) space and that is discrete and acts properly discontinuously on X. If is a Holder -quasiconformal measure, then there is a measure on such that (X, )is the Poisson boundary for (, ).</p><p>The most important examples to which we apply this theorem is givenby the following result which we prove in section 9. (See section 3.2 fordenition of a -conformal metric.)</p><p>Corollary 0.4. If X, are as in the above theorem and is uniformlyHolder equivalent to a PattersonSullivan measure, or equivalently, thecritical Hausdor measure of a -conformal metric, then there is a measure on such that (X, ) is the Poisson boundary for (, ).</p><p>Remark 0.5. In each of the results above, we nd solution measures which are an innite sum of point measures. However in Corollary 9.4 weshow that, in each case, the family of stationizing measures is innitedimensional and has members in any Borel measure class supported on allof . (In Theorem 0.3 there is only one such class since is discrete.)</p></li><li><p>Vol. 17, 2007 HARMONIC MEASURES 713</p><p>So far, we are unable to determine general criteria for when a symmetricexample always exists; i.e. one with () = (1) for all .</p><p>Before leaving the introduction, we point out to the reader one of thesimplest examples to which these results apply.Example 0.6. Let be a nitely generated free group and X be theCayley graph (a uniform tree) of with respect to any nite generatingset. Let = p be the PattersonSullivan measure of any -invariantCAT(-1) metric on X. For any positive Holder function, f : X R, Theprevious corollary shows that (X, f) is the Poisson boundary for somerandom walk (, ). We will explore this example in a little more detail insection 10.</p><p>We conclude with a brief outline of the paper. In section 1 we introducedenitions and some basic tools used in working with nongeodesic Gro-mov hyperbolic spaces before presenting our main results in their generalform. In section 2, we develop some consequences of a generalized shadowlemma which will be necessary later. Section 3 develops certain aspects ofPattersonSullivan theory in the context of CAT(1) spaces. Sections 4and 5 present the notation and background for the conditions which willbe needed in order to guarantee that a family of functions can form apositive basis. In section 6 we present a general theorem of independentinterest which establishes when lower semicontinuous functions can be ap-proximated by positive sums of basis functions. In sec...</p></li></ul>