ERGODIC PROPERTIES OF THE HOROCYCLE FLOW AND CLASSIFICATION OF FUCHSIAN GROUPS. Vadim A. Kaimanovich. September 1999

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1 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW AND CLASSIFICATION OF FUCHSIAN GROUPS Vadim A. Kaimanovich September 1999 Abstract. The paper is devoted to a study of the basic ergodic properties (ergodicity and conservativity) of the horocycle ow on surfaces of constant negative curvature with respect to the Liouville invariant measure. We give several criteria for ergodicity and conservativity and connect them with the classication of the associated Fuchsian groups. Special attention is given to covering surfaces. In particular, we show that normal subgroups of divergent type Fuchsian groups provide natural examples for the strictness of a number of inclusions in the classication of Fuchsian groups. The paper in a sense complements and continues earlier work by A. N. Starkov \Fuchsian groups from the dynamical viewpoint" published in J. Dynam. Control Systems in Introduction The horocycle and the geodesic ows on surfaces of constant negative curvature are the basic examples of ows on homogeneous spaces. The general theory of such ows so far mostly deals with the conite volume case (e.g., see [St99]). In this situation establishing ergodicity (conservativity being self-evident) is just the rst step before studying much more subtle properties. In particular, ergodicity of both the horocycle and the geodesic ows with respect to the Liouville measure on nite volume surfaces has been known for a long time. In the innite volume case even such basic ergodic properties as ergodicity and conservativity of the Liouville measure become non-trivial. For the geodesic ow this problem is completely resolved by the Hopf{Tsuji{Sullivan theorem [Su81], according to which ergodicity is equivalent to conservativity of the geodesic ow, and both are equivalent to recurrence of the Brownian motion on the surface ( the associated Fuchsian group being of divergent type). However, until recently little was known about the ergodic properties of the horocycle ow in the innite volume case. The paper [St95] by Starkov resuscitated this problem. In particular, he conjectured that ergodicity of the horocycle ow is equivalent to ergodicity of the boundary action of the associated Fuchsian group. Inspired by this conjecture, Babillot and Ledrappier [BL98] proved that the horocycle ow is ergodic on all Z d -covers of compact surfaces (independently of the degree of the cover, in contrast to the geodesic ow case); another proof of this fact was later given by Pollicott [Po98]. Here we prove Starkov's conjecture in full generality. In fact, we give a complete description of the ergodic components of the horocycle ow in terms of 1991 Mathematics Subject Classication. Primary 20H10, 58F17; Secondary 28D15. Typeset by AMS-TEX

2 2 VADIM A. KAIMANOVICH the ergodic decomposition of the associated boundary action (Theorem 4.2). This proof is at the origin of the present paper. The study of the ergodic properties of the horocycle ow lies at the crossroads of several subject areas (e.g., ergodic theory of general groups of transformations with quasi-invariant measure, Brownian motion and harmonic functions on manifolds, classication of Fuchsian groups, etc.). Although at rst glance some of these subjects may look rather distant from the horocycle ow, actually they often provide prompt answers to questions connected with the horocycle ow. Because of this we decided to give the paper a more expository character by discussing some relevant results and methods from these areas as well. Sections 1{4 contain background denitions and results. Although they are certainly known to specialists (actually, for some of them it would be dicult to nd a precise reference), we included them in order to make the paper more self-contained and accessible, so that it might also serve as an introduction to the subject. In Sections 1 and 2 we discuss several models of the hyperbolic plane and give several interpretations of the Busemann cocycle on the hyperbolic plane. In particular, the fact that the Busemann cocycle is proportional to the Radon{Nikodym cocycle of the family of visibility measures on the boundary circle (Lemma 2.2) plays the central role in the proof of Theorem 4.2. In Section 3 we give several descriptions of the action of the group Iso(H 2 ) = SL(2; R) on the space of horocycles Hor(H 2 ) (Lemma 3.2). Finally, in Section 4 we dene the horocycle ow on a surface M = GnH 2, where G is a Fuchsian group, and show that the ergodic properties of the horocycle ow and of the action of G on Hor(H 2 ) are the same. Section 5 is devoted to a proof of Theorem 4.2. It is based on the proportionality of the Busemann and Radon{Nikodym cocycles. Therefore, the ergodic decomposition of the horocycle ow on M is the same as the ergodic decomposition of the Radon{ Nikodym extension of the boundary action of the group G. In turn, by the general theory the Radon{Nikodym extension of an arbitrary measure type preserving action is ergodic i the original action is ergodic and its ratio set is R +. The Radon{Nikodym extension of a conservative action is also conservative (once again, a general property). As a consequence, the Radon{Nikodym cocycle takes values arbitrarily close to 0 on any positive measure subset of the conservative part of the boundary action. A Lebesgue density theorem type argument then implies that all positive real numbers must belong to the ratio set. The latter argument follows the outlines of the proof of Sullivan [Su82] (see also Sullivan [Su81]) of the fact that the boundary action of a Kleinian group is of type III 1 on its conservative part. We begin Section 6 with a discussion of the boundary decomposition of positive harmonic functions on Riemannian manifolds, which is used to obtain a characterization of the conservativity set of the boundary action ( of the horocycle ow) as the divergence set of the \Poincare{Busemann series" (Proposition 6.3 and Theorem 6.5). In view of the Poisson formula, ergodicity of the boundary action of G ( ergodicity of the horocycle ow) is equivalent to absence of bounded harmonic functions on the quotient surface M (Theorem 6.4). Although for the horocycle ow ergodicity and conservativity and not equivalent (unlike for the geodesic ow), these two results taken together bear a strong similarity to the Hopf{Tsuji{Sullivan theorem. In Section 7 we consider the situation when a surface M is a regular cover of a

3 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 3 \smaller" surface M. For the corresponding Fuchsian groups G and G it means that G is a normal subgroup of G. In the case when the group G is of divergent type (equivalently, the geodesic ow on M is ergodic, or the Brownian motion on M is recurrent) we give another simpler proof of Theorem 4.2. Combining Theorem 6.4 with earlier results on absence of bounded harmonic functions on covering manifolds we obtain several ecient conditions for ergodicity of the horocycle ow (Theorem 7.2); in particular, it is ergodic for polycyclic covers of compact surfaces and for nilpotent covers of recurrent surfaces. Further, we prove that the horocycle ow on a regular cover of a recurrent surface is always conservative, and that it is either ergodic or all its ergodic components have measure 0 (Theorem 7.7). In view of Theorem 4.2, Theorem 7.7 is equivalent to an analogous claim for the boundary action of a normal subgroup of a divergent type Fuchsian group (Theorem 7.3). The latter was rst proved by Velling and Matsuzaki [VM91]. We give another proof of Theorem 7.3 which is based on the theory of covering Markov operators from [Ka95] and uses just the coincidence of the visibility and harmonic measures in the constant curvature case. In virtue of the results from Section 7 normal subgroups of divergent type Fuchsian groups provide natural examples for the strictness of a number of inclusions in the classication of Fuchsian groups (earlier examples of this kind were obtained by analytic constructions). These examples are discussed in Section 8. Although for the sake of brevity we consider here the horocycle ow only, our results and proofs verbatim carry over to the higher-dimensional situation and give a description of the ergodic properties of the horosphere foliation on an arbitrary manifold of constant negative curvature (this is why we often prefer to use the \higher-dimensional terms": call the boundary 2 a sphere, talk about balls and annuli 2, etc.). 1. The hyperbolic plane Let H 2 be the hyperbolic plane with the Riemannian metric d, i.e., the unique (up to isometries) 2-dimensional simply connected Riemannian manifold with sectional curvature?1. Denote 2 its boundary circle (the set of all asymptotic classes of geodesic rays), which is the boundary of the visibility compactication of H 2 : an (escaping to innity) sequence of points x n 2 H 2 converges in this compactication i for a certain ( any) reference point o 2 H 2 the directing vectors of the geodesic segments [o; x n ] converge. 2 is identied with the unit circle of the tangent space at o via the Riemannian exponential map at o. The action of the group Iso(H 2 ) of isometries of H 2 extends to a continuous action on the boundary 2. See [BGS85] for a discussion of the visibility compactication of general Cartan{Hadamard manifolds. In the disk (Poincare) model of the hyperbolic 2 is indeed the boundary circle, whereas in the upper half-plane (Lobachevsky) model it is the union of the boundary line and the point at innity. In both these models the visibility topology coincides with the \usual" topology (the one induced from the embedding of the corresponding model into R 2 or its one-point compactication, respectively). Below we prefer to use the upper half-plane model, because the computations there are easier. The hyperbolic metric in the upper half-plane model H 2 = R 2 + = f(x; y) 2 R 2 : y > 0g

4 4 VADIM A. KAIMANOVICH has the form and the Riemannian volume is ds 2 = dx2 + dy 2 y 2 ; d(x; y) = dx dy y 2 : The connected component Iso 0 (H 2 ) of the isometry group Iso(H 2 ) is isomorphic to the group P SL(2; R) acting on R 2 + C by complex fractional linear transformations a b z = az + b c d cz + d ; z = x + iy : Obviously, the action of Iso 0 (H 2 ) on H 2 is transitive. The full isometry group Iso(H 2 ) = SL(2; R) is a Z 2 -extension of Iso 0 (H 2 ) generated by the symmetry (x; y) 7! (?x; y). The stabilizer of the point i = (0; 1) is (1.1) Stab(i) = SO(2) SL(2; R) = Iso(H 2 ) : In the ellipses model (provided by the general theory of symmetric spaces) the hyperbolic plane is identied with the space of all area ellipses in R 2 centered at the origin. Indeed, by (1.1) H 2 = SL(2; R)=SO(2), which allows one to identify points x 2 H2 with their stabilizer subgroups Stab(x) SL(2; R). Being a conjugate of SO(2), any subgroup Stab(x) is the group of rotations of a certain Euclidean structure " x on R 2 ; then the associated ellipse E x is the unit circle of " x. Obviously, Stab(gx) = gstab(x)g?1, and E gx = ge x. Since for the reference point o = i 2 H 2 in the upper half-plane model Stab(o) = SO(2), the corresponding ellipse E o is just the unit circle of the standard Euclidean metric on R 2. Lemma 1.1. In the ellipses model d(x; y) = 2 log kk x!y ; where kk x!y = sup v kvk y kvk x is the operator norm of the identity map on R 2 from the Euclidean structure " x to the Euclidean structure " y, i.e., the quasi-conformal deformation between " x and " y. Proof. Look at the upper half-plane model. Since the only isometry invariant of pairs of points in H 2 is the distance, without loss of generality we may assume that x = i and y = gi = 2 0 i with g = 0 1=, where > 1. Then d(i; gi) = 2 log. On the other hand, the ellipse E x is the unite circle, and the ellipse E y = ge x has semi-axes and 1=, so that kk x!y =. In view of Lemma 1.1 the geodesic rays (x t ) (with the natural parameterization) of the hyperbolic metric on H 2 admit a simple description in the ellipses model. Namely,

5 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 5 take the Euclidean structure " = " x0 on R 2 corresponding to the starting point of the ray. Then the big semi-axes of the ellipses E xt all have the same direction and their length is e t=2. In order to obtain the negative ray of the same bilateral geodesic one has to exchange the big and the small semi-axes (the direction of the former big semi-axis becomes contracting instead of expanding), so that the angle between the big semi-axes of the positive and the negative rays of the same bilateral geodesic is =2. More generally, the angle between two geodesic rays issued from the same point is twice the angle between the directions of the big semi-axes of the corresponding families of ellipses. A sequence of ellipses converges in the visibility compactication i their aspect ratio tends to innity and the directions of their big axes converge, so 2 can be identied with the projective line RP (we identify points 2 with straight lines in R 2 passing through the origin). This identication conjugates the action of Iso(H 2 ) = SL(2; R) 2 with its standard action on RP (the quotient of the linear action on R 2 : = R 2 n f0g). 2. The Busemann cocycle For any x; y 2 H 2 ; 2 let (2.1) (x; y) = lim z n! d(y; zn )? d(x; z n ) ; where the sequence z n 2 H 2 converges to in the visibility compactication. The limit (2.1) exists and is independent of the choice of the sequence z n (e.g., see [BGS85]). In other words, (x; y) can be considered as a \regularization" of the formal expression d(y; )? d(x; ) with being a point at innity. For any xed 2 the function satises the (additive) cocycle identity (x; y) + (y; z) = (x; z) 8 x; y; z 2 H 2 ; and we shall call the Busemann cocycle on H 2 associated with the boundary point. Each of the cocycles is cohomologically trivial 1. However, since the whole family of Busemann cocycles is isometry invariant, i.e., (x; y) = g (gx; gy) 8 x; y 2 H 2 ; 2 ; g 2 Iso(H 2 ) : it gives rise to a non-trivial cocycle of the action of the group Iso(H 2 ) 2 (see below Section 3). The level set Hor (x) = fz 2 H 2 : (x; z) = 0g 1 The Busemann cocycle rst introduced in [Ka90] and used in a number of later papers had the opposite sign. The choice of sign in the present paper is more consistent with other notations. In particular, the prefer to write the coboundary ( the incremental cocycle) associated with a function ' as c ' (x; y) = '(y)?'(x). Then under our present convention the Busemann cocycle is the coboundary associated with the Busemann function x 7! (o; x) determined by a xed reference point o 2 H 2.

6 6 VADIM A. KAIMANOVICH is called the horocycle centered at the point 2 and passing through the point x 2 H 2. For any sequence z n! the Riemannian spheres S zn (x) centered at z n and passing through x converge pointwise to the horocycle Hor (x). Denote by Hor(H 2 ) the space of all horocycles in H 2. In the disk or the upper half-plane models of the hyperbolic plane horocycles are Euclidean circles tangent to the boundary circle (resp., to the boundary line). The horocycles centered at the point at innity in the upper half-plane model are horizontal lines, and the corresponding Busemann cocycle has the form (2.2)? (x1 ; y 1 ); (x 2 ; y 2 ) = log y 1 y 2 : Lemma 2.1. Any measurable Iso(H 2 )-invariant family of cocycles on H 2 indexed 2 is a multiple of the family of Busemann cocycles. Proof. Let fc g; 2 be such a family of cocycles. For any x; y 2 H 2 ; 2 with (x; y) = 0 there is an isometry g 2 Iso(H 2 ) which xes and swaps x and y (take for the point at innity in the upper half-plane model and for g the axial symmetry sending x to y). Therefore, if (x; y) = 0, then c (x; y) = 0. It means that for calculating the values of c we only have to consider points (x; y; ) such that is the endpoint of the geodesic segment [x; y]. Since the distance is the only isometry invariant on the space of pairs of points from H 2, we conclude that c must be an additive function of, i.e., to be proportional to. We shall now give several other interpretations of the family of Busemann cocycles. In the ellipses model let (x; y) = log kk y kk x be the norm cocycle obtained by taking the logarithm of the ratio of the Euclidean norms k k x ; k k y ; x; y 2 H 2 of vectors representing a point 2 = RP. Given a point x 2 H 2 denote by m x the unique probability measure 2 invariant with respect to all isometries of H 2 xing the point x. The measure m x is the image of the normalized Lebesgue measure on the sphere of the tangent space at x under the exponential map and it is called the visibility measure with the pole at x. Obviously, gm x = m gx for any isometry g 2 Iso(H 2 ). The measures m x ; x 2 X are all pairwise equivalent, so that for any 2 we have the Radon{Nikodym cocycle (x; y) = log dm y dm x () : Lemma 2.2. For any x; y 2 H 2 and 2 (x; y) = 2 (x; y) =? (x; y) : Proof. Since the cocycles and are both isometry invariant, by Lemma 2.1 we just have to nd the corresponding proportionality coecients.

7 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 7 Let us rst look at the cocycle. By Lemma 1.1 the absolute value of the proportionality coecient is 2, and we only have to choose the sign. Take the same points x = i and y = 2 i 2 H 2 in the upper half-plane model as in the proof of Lemma 1.1, and let 2 be the endpoint of the geodesic ray issued from x in the direction of y. Then (x; y) =?d(x; y) =?2 log. On the other hand, in the ellipses model the point corresponds to the big axis of the ellipse E y, i.e., to the vector v = (1; 0). Then kvk y =kvk x = 1=, and we obtain = 2. For identifying the cocycle we need to nd the Radon{Nikodym derivatives of the visibility measures. In the ellipses model the measure m x ; x 2 H 2 2 = RP is the image of the unique Stab(x)-invariant measure on E x under the projective factorization 0 =?. Take again the same points x; y;. Then m y = 0 1= calculation shows that dm y =dm x () = 2, whence =?. m x, and an elementary Remarks. 1. One can also compare the cocycles and directly. Take two points x and y = gx from H 2. Since the Lebesgue measure on R 2 is g-invariant, passing to the polar coordinates we have r dr d = r 0 dr 0 d 0, where (r 0 ; 0 ) = g(r; ). r=1;=' = r r In other words, for the point = f 0 ;? 0 g dm y dm x () = dgm x dm x @r 0 r=1;=' = 1 r 0 2 :!?1 = (r 0 ) 2 = r=1;=' kkx kk y 2 2. For yet another interpretation of the Busemann cocycles we remind that for any boundary point 2 and any reference point o 2 H 2 the function ' o : x 7! e (x;o) = dm x dm o () is a minimal harmonic function on H 2 (i.e., any harmonic function f with 0 f ' o is proportional to ' o), and all minimal harmonic functions have this form (in fact, in the disk model (o; x; ) = dm x =dm o () is precisely the classical Poisson kernel, cf. Section 6 below). Thus, the 2 can be identied with the space of minimal harmonic rays ', i.e., with the minimal Martin boundary of the hyperbolic plane (in our case it actually coincides with the full Martin boundary); the function ' o is the representative of the ray ' normalized by the condition ' o (o) = 1. Then the ratio (x) is obviously independent of the point o, and its logarithm (which in our ' o (y)=' o case coincides with? (x; y)) can be called the harmonic cocycle.

8 8 VADIM A. KAIMANOVICH 3. The boundary action Since the horocycles are dened in entirely metric terms, ghor (x) = Hor g (gx) for any x 2 H 2 ; 2 and g 2 Iso(H 2 ). We shall now give an explicit formula for this action in terms of the Busemann cocycle. Assigning to horocycles their centers we obtain an R-bration : Hor(H 2 2. Any section 2! Hor(H 2 ) determines a trivialization : Hor(H 2 )! R of by the formula (3.1) () = ()? ; () ; i.e., () is the signed distance between the concentric horocycles and (). Then for the horocycle g; g 2 Iso(H 2 ) we have (g) = (g)? g; (g) = ()? ; g?1 [ (g)] = ()? ; (g?1 ) () ; where (g?1 )() = g?1 [(g)] is the g?1 -translation of the section, whence (3.2) (g)? () = ()? ; (g?1 ) ()? ()? ; () = ()? (); (g?1 ) () = b? g; () is the signed distance between the sections and g?1 at the point (). Therefore, the action of Iso(H 2 ) on Hor(H 2 ) 2 R in terms of the trivialization (3.1) takes the form (3.3) g(; t) =? g; t + b (g; ) : Associativity of the action implies that the additional term c = b appearing in the right-hand side of (3.3) satises the cocycle identity (for group actions) (3.4) c(g 1 g 2 ; ) = c(g 1 ; g 2 ) + c(g 2 ; ) 8 g 1 ; g 2 2 Iso(H 2 ); 2 : Conversely, any R-valued cocycle c of the action of Iso(H 2 ) 2 (i.e., a function c : Iso(H 2 2! R satisfying the identity (3.4)) determines the corresponding skew action of the group Iso(R 2 ) 2 R (3.5) g(; t) =? g; t + c(g; ) : Moreover, by formula (3.2) c uniquely (up to an additive constant) determines the associated trivialization (i.e., the section ). If 0 2! Iso(H 2 ) is another section, then by formula (3.2) b 0(g; )? b (g; ) =? 0 (); ()? g? 0 (g); (g) ; so that the cocycles b 0 and b are cohomologous. We shall call their cohomology class the Busemann class. Choosing a reference point o 2 H 2 gives a section o () = Hor (o) of the ber bundle : Hor(H 2 2. The associated trivialization o = o (3.1) and the cocycle b o = b o (3.2) then take the form and we have o? Hor (x) = (x; o) ; b o (g; ) = (o; g?1 o) ;

9 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 9 Lemma 3.1. The action of the group Iso(H 2 ) on the space Hor(H 2 ) is conjugate with the skew action of Iso(H 2 ) 2 R determined by the cocycle b o (g; ) = (o; g?1 o), where o 2 H 2 is a reference point. We shall now use Lemma 2.2 in order to give other equivalent descriptions of the action of the group Iso(H 2 ) on the space of horocycles. At the end of Section 1 we have identied points of the boundary 2 with the projective line RP. Under this identication the action of Iso(H 2 ) = SL(2; R) 2 is conjugate with the standard action of SL(2; R) on RP (the quotient of the linear action on R 2 : = R 2 n f0g). Denote by R 2 : = the set of all non-zero vectors from R 2 factorized by the equivalence relation fv;?vg. Then the projection fv;?vg 7! = Rv from R 2 : = to RP is an R-bration. Taking a norm k k on R 2, we obtain a trivialization : fv;?vg 7! log kvk of this bration. Therefore (cf. formula (3.2)), the action of SL(2; R) on R 2 : = is conjugate to the skew action on RP R determined by the cocycle n(g; ) = log kgk=kk (since the ratio of norms depends only on the projective class of a vector v 2 R 2, in this formula we replace the vector v with its projective class = Rv 2 RP ). For the norm k k o determined by a reference point o 2 H 2 in the ellipses model this cocycle takes the form n o (g; ) = log kgk o kk o = log kk g?1 o kk o = (o; g?1 o) : Recall that any measure type preserving action of a group G on a measure space (X; m) determines the measurable Radon{Nikodym cocycle (3.6) r(g; x) = log dg?1 m dm (x) : We shall call the associated skew action (3.5) on X R determined by the Radon{ Nikodym cocycle (3.7) g(x; t) = (gx; t + log dg?1 m dm (x)) the Radon{Nikodym action, or the Radon{Nikodym extension of the original action on X. In the case of the group G = Zthis action is also known under the name of the Maharam transformation, see [Aa97]. Below we shall always endow the space X R of the Radon{Nikodym action with (the type of) the quasi-invariant measure m, where is the Lebesgue measure on R. The Radon{Nikodym cocycles corresponding to two equivalent measures on the same G-space are cohomologous, and the associated Radon{Nikodym actions are conjugate. Fix a reference point o 2 H 2 and the corresponding visibility measure m o 2. Then the associated Radon{Nikodym cocycle of the action of Iso(H 2 ) 2 takes the form (3.8) r o (g; ) = log dg?1 m o () = log dm g?1 o () = (o; g?1 o) : dm o dm o Combining Lemma 3.1 with the equivalent descriptions of the Busemann cocycle from Lemma 2.2 and Remark 2 after it we now obtain

10 10 VADIM A. KAIMANOVICH Lemma 3.2. The following actions of the group Iso(H 2 ) = SL(2; R) are all pairwise conjugate: (i) the action of Iso(H 2 ) on Hor(H 2 ); (ii) the Radon{Nikodym action of Iso(H 2 ) 2 R; (iii) the linear action of SL(2; R) on R 2 : =; (iv) the action of Iso(H 2 ) on the space of minimal harmonic functions on H 2. Remark. The conjugacy of the actions on Hor(H 2 ) and on R 2 : = can be also veried in a more direct (if much more tedious) way. Begin with an elementary geometric formula. The circle on the upper half plane tangent to the horizontal axis at the point (; 0) and passing through a point (x; y) crosses the vertical line starting from (; 0) at the point (; h) with h = ((x? ) 2 + y 2 )=y (for, the distance between (; h=2) and (x; y) has to be h=2). Thus, in the upper half-plane model, for the reference point o = i = (0; 1), we obtain? y( 2 + 1) (x; y); o = log where 2 R. Now, for a matrix g = a b c d go = x + iy = ai + b ci + d (x? ) 2 + y 2 ; ac + bd? bc = + iad c 2 + d2 c 2 + d 2 2 SL(2; R) we have in complex coordinates ac + bd = c 2 + d + i 1 2 c 2 + d ; 2 so that x = (ac + bd)=(c 2 + d 2 ) and y = 1=(c 2 + d 2 ). Since g = (a + b)=(c + d), we have b o (g; ) = (o; g?1 o) = g (go; o) = log = log 1 c 2 +d [( a+b 2 c+d )2 + 1] ( ac+bd c 2 +d? a+b 2 c+d )2 1 + (c 2 +d 2 ) 2 (c 2 + d 2 )[(a + b) 2 + (c + d) 2 ] [(ac + bd)(c + d)? (c 2 + d 2 )(a + b)] 2 + (c + d) 2 = log (c2 + d 2 )[(a + b) 2 + (c + d) 2 ] (c? d) 2 + (c + d) 2 = log (a + b)2 + (c + d) k(a + b; c + d)k = 2 log = 2 log k(; 1)k kg(; 1)k k(; 1)k = 2n o(g; ) : Below we shall be working with the measure category, and all actions (unless otherwise specied) will be considered with respect to the natural smooth measure type of the corresponding spaces.

11 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW The horocycle flow Denote by UH 2 the unitary tangent bundle of the hyperbolic plane (the space of unit length tangent vectors). Then any vector v 2 UH 2 is tangent to two horocycles passing through its base point; the centers of these horocycles are the endpoints of the geodesic perpendicular to v. The fact that H 2 is oriented allows one to choose one of these horocycles in the same manner for all vectors v (for example, in such a way that the vector v is directed anticlockwise along the horocycle). Denote this horocycle by (v). The horocycle ow fh t g then consists in moving vectors v 2 UH 2 along the associated horocycles (v) with unit speed. The action of the connected component Iso 0 (H 2 ) of the group of isometries of H 2 on UH 2 is simply transitive (for, the isometries preserving orientation and xing a point o 2 H 2 are just the rotations around o). Therefore, from the group theoretical point of view one can identify the space UH 2 with Iso 0 (H 2 ) = P SL(2; R). In the upper half-plane model let v 0 2 UH 2 be the unit length vector with the basepoint i = (0; 1) parallel to the horizontal axis and pointed at the positive direction. The horocycle (v 0 ) is then the horizontal line passing through i, so that taking into account the. Since the group Iso 0 (H 2 ) acts on choice of the orientation we have H t v 0 = UH 2 on the left, we obtain that nthe horocycle o ow is conjugate with the right action of the unipotent subgroup U = on P SL(2; R). In particular, since this action 1 t 0 1 preserves the Haar measure on P SL(2; R), the horocycle ow preserves the Liouville measure on UH 2 (which is the image of the Haar measure). See Lemma 5.1 below for another explanation of this fact. Let G be a Fuchsian group, i.e., a discrete subgroup of Iso 0 (H 2 ) = P SL(2; R). Denote by M = GnH 2 the corresponding quotient surface (strictly speaking, M is a smooth surface only if G contains no elliptic elements; otherwise M is an orbifold and has singular points). Since G preserves the orientation of H 2, the ow H t descends from UH 2 to the horocycle ow on UM = GnUH 2, which we shall also denote by H t. From the group theoretical point of view we have here two commuting actions on UH 2 = Iso0 (H 2 ): the left action by the group of isometries G and the right action by the unipotent subgroup U. The quotient Iso 0 =U is the space of trajectories of the horocycle ow on H 2, i.e., the space Hor(H 2 ) of horocycles in H 2. Since both groups G and U are unimodular and closed, we have (see [Fu73, Theorem 1.4]) Lemma 4.1. Given a Fuchsian group G and the associated quotient surface M = GnH 2, there exists a natural convex isomorphism between the following 3 spaces of Radon measures: (1) The space of G-invariant measures on Hor(H 2 ); (2) The space of H t -invariant measures on UM; (3) The space of simultaneously G- and H t -invariant measures on UH 2. 1 t 0 1 Corollary. Given a Fuchsian group G, the horocycle ow on UM = GnUH 2 is ergodic (resp., conservative) i the action of G on the space Hor(H 2 ) is ergodic (resp., conservative).

12 12 VADIM A. KAIMANOVICH In view of Lemma 4.1 the following Theorem gives a complete description of the ergodic components of the horocycle ow on UM = GnUH 2 in terms of the ergodic components of the action of a Fuchsian group G on the boundary 2 (recall the notation : Hor(H 2 2 for the bration of the space of horocycles over the boundary circle introduced in Section 3). Theorem 4.2. Let G be a Fuchsian group, 2 = C [ D be the decomposition of the boundary circle into the conservative and the dissipative parts of the G-action. Then the conservative and the dissipative parts of the action of G on the space Hor(H 2 ) are?1 (C) and?1 (D), respectively. The conservative ergodic components of the G- action on Hor(H 2 ) are?1 (C ), where C = S C is the ergodic decomposition of the G-action on C. Corollary 1. The horocycle ow on GnUH 2 is ergodic i the action of G 2 is ergodic. In view of Lemma 3.2 we have Corollary 2. The linear action of a Fuchsian group G on R 2 : = is ergodic i its action on RP is ergodic. Since the action of G on R 2 : (resp., on S 1 ) is a 2-fold cover of the action on R 2 : = (resp., on RP = S 1 =), we also have the following result, which, in spite of its surprisingly elementary formulation, is apparently new. Corollary 3. The linear action of a Fuchsian group G on R 2 is ergodic i its action on S 1 is ergodic. The result of Corollary 1 was conjectured by Starkov [St95]. In the particular case when M is an abelian cover of a compact surface it was proved by Babillot and Ledrappier [BL98] and yet another proof was later given by Pollicott [Po98]. The next Section is entirely devoted to a proof of Theorem Ergodicity of the Radon{Nikodym cocycle We begin with recalling some properties of the Radon{Nikodym cocycle (3.6) of a general measure type preserving action of a countable group G on a measure space (X; m). The rst one is straightforward Proposition 5.1. The measure d(x; t) = e?t dm(x)dt on X R is preserved by the Radon{Nikodym action (3.7). The second property is also well-known (see [Sc77]). However, the proof that we give here is dierent and much simpler.

13 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 13 Proposition 5.2. The conservativity of the action of G on (X; m) is equivalent to conservativity of its Radon{Nikodym extension. Proof. Replacing, if necessary, the measure m with an equivalent one, we may always assume that m(x) = 1. Let be the corresponding G-invariant measure of the Radon{ Nikodym action (see Proposition 5.1). Then (X R + ) = 1. Suppose that the Radon{Nikodym action has a non-trivial dissipative part, i.e., that there exists a G-invariant subset D X R with (D) 6= 0 such that the ergodic components of the restriction of the Radon{Nikodym action to D are G-orbits (we call such orbits dissipative). Since (XR + ) = 1, and the Radon{Nikodym action preserves the measure, a.e. dissipative orbit G(x; t) = f(x i ; t i )g intersects X R + only nitely many times, so that = (x; t) = max i t i < 1, which implies that the Radon{Nikodym cocycle must be cohomologically trivial. Indeed, by denition, the function on X R is G-invariant, and (x; t + t 0 ) = (x; t) + t 0 for any t 0 2 R. Therefore, if we put '(x) = (x; t)? t = (x; 0), then formula (3.7) implies that '(x) = (x; t)? t = (gx; t + r(g; x))? t = '(gx) + r(g; x) ; so that log dg?1 m=dm(x) = r(g; x) = '(x)? '(gx). Thus, the measure dm 0 (x) = e '(x) dm(x) on X is G-invariant. Since the Radon{ Nikodym action determined by the measure m is conjugate to the Radon{Nikodym action determined by the measure m 0, the problem is therefore reduced to the trivial case when the measure m is G-invariant. In the latter case the Radon{Nikodym cocycle is identically zero, so that the Radon{Nikodym action on each of the layers X ftg coincides with the original action, which is conservative by our hypothesis. Denition. Let T be a measure type preserving action of a countable group G on a Lebesgue measure space (X; m). The ratio set R(T ) of this action is the set of all numbers t > 0 which can be approximated by the Radon{Nikodym derivatives of the action restricted to an arbitrary subset of X. Namely, t 2 R(T ) if for any subset Z X with m(z) > 0 and any " > 0 there exist g 2 G and a subset A Z with m(a) > 0 such that ga Z and the Radon{Nikodym derivative of the transformation g restricted to A is in the interval [t? "; t + "]. Put (5.1) (g; I) = fx 2 X : log dgm (x) 2 Ig ; g 2 G; I R : dm Then the above denition means that t 2 R(T ) if for any Z X with m(z) > 0 and any interval I containing t in its interior there exists g 2 G such that the set has non-zero measure m. Z \ gz \ (g; I)

14 14 VADIM A. KAIMANOVICH Proposition 5.3 [Sc77], [HO81]. The Radon{Nikodym action of the group G determined by an ergodic action T is ergodic i R(T ) = R +. Now, in view of Lemma 3.2 and Lemma 4.1, Theorem 4.2 would follow from Proposition 5.4. Let T be the canonical action of a Fuchsian group on the limit circle S 1 2 with a non-trivial conservative part 2. Then R(T C ) = R +, where T C is the restriction of the action T onto C. Before proving Proposition 5.4, we shall need a couple of auxiliary statements. Below in this Section we shall x a reference point o 2 H 2 and denote by m = m o and = o the corresponding visibility measure and the visibility metric (the image of the standard metric on the unit circle of the tangent space under the exponential map) 2, respectively. For notational simplicity we shall also put m(a) = jaj for measurable subsets 2. First note that in view of Lemma 2.2 and formula (3.8) the sets (g; I) can be also dened as (g; I) = f 2 : (go; o) 2 Ig : Then the hyperbolic cosine rule (e.g., see [Be95]) easily implies Lemma 5.5. If jtj < d(o; x), then (g; [t; 1)) is a ball of the metric 2 centered at the endpoint of the geodesic ray issued from o in the direction of go. Its radius R(g; t) depends on the hyperbolic distance d = d(o; go) and t only, and for any xed t R(g; t) 2e?(t+d)=2 ; where means that the ratio of both sides tends to 1 as d! 1. Denition (cf. [Gu76]). We shall say that a family of measurable sets 2 is a density family if for any measurable 2 with jzj > 0 and a.e. 2 jz \ Aj inf : A 2 fa g; x 2 A; diam A < r?! 1 : jaj r!0 Lemma 5.6. For any interval I R the family of sets (g; 2 ; g 2 Iso(H 2 ) is a density family. Proof. According to Lemma 5.5 the set (g; I) is a metric annulus 2 (provided d(o; go) is suciently big). Denote by B(g; I) its outer ball. By Lemma 5.5 diam B(g; I)! 0 as d(o; go)! 1, and the ratio of the outer and inner radii of (g; I) converges to a non-zero limit. Therefore, the limit of the ratio j(g; I)j=jB(g; I)j exists and is non-zero. Since the family of metric balls 2 is a density family by the classical Lebesgue density theorem, the family f(g; I)g is also a density family.

15 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 15 Corollary. For any interval I R, a subset 2 with jzj > 0 and " > 0 there exist a number r 0 = r 0 (I; Z; ") > 0 and a subset Z 0 = Z 0 (I; Z; ") Z with jz 0 j > 0 such that (5.2) jz \ (g; I)j j(g; I)j > 1? " whenever diam (g; I) < r 0 and Z 0 \ (g; I) 6=?. Now we are ready to nish the proof of Theorem 4.2 Proof of Proposition 5.4. We have to prove that for any subset Z C with jzj > 0 and any interval I R there exists g 2 G such that (5.3) jz \ gz \ (g; I)j > 0 : Let I 0 = [?T; T ] R be a symmetric interval containing I. Fix a small number " > 0 (its value to be specied later), and let Z 0 = Z 0 (I 0 ; Z; ") and r 0 = r 0 (I 0 ; Z; "). By Proposition 5.2 we know that the Radon{Nikodym action on C R is conservative. Therefore, there exists an innite number of group elements g n 2 G such that (5.4) jz 0 \ g n Z 0 \ (g n ; I 0 )j > 0 : Since G is a Fuchsian group, d(o; g n o)! 1, so that by Lemma 5.5 (5.5) diam (g n ; I 0 )! 0 ; and there exists a limit (5.6) lim j(g n; I)j j(g n ; I 0 )j = > 0 : We shall now deduce (5.3) from (5.4) by using Lemma 5.6 and its Corollary. The argument is straightforward. Indeed, by (5.2), (5.4) and (5.5) jz \ (g n ; I 0 )j j(g n ; I 0 )j > 1? " for all suciently big n. Obviously, so that from (5.6) jz \ (g; I)j jz \ (g; I 0 )j? j(g; I 0 ) n (g; I)j ; (5.7) lim inf jz \ (g n; I)j j(g n ; I)j > 1? " : Now we have to estimate jg n Z \ (g n ; I)j = jg n (Z \ g?1 n (g n ; I))j :

16 16 VADIM A. KAIMANOVICH Since g?1 (g; I) = f : g (go; o) 2 Ig = f : (o; g?1 o) 2 Ig = f : (g?1 o; o) 2?Ig = (g?1 ;?I) ; in the same way as above we have lim inf jz \ g?1 n (g n ; I)j jg n?1 (g n ; I)j > 1? " ; where = lim j(g n ;?I)j=j(g n ; I 0 )j. Now, since I I 0 = [?T; T ], by the denition of the sets (g n ; I) (5.1) we have that the Radon{Nikodym derivatives of the transformations g n restricted to g n?1 (g n ; I) = (g n?1 ;?I) are between e?t and e T, so that (5.8) lim inf jg nz \ (g n ; I)j j(g n ; I)j > 1? e 2T " ; Therefore, if " is chosen in such way that the right-hand sides in the inequalities (5.7) and (5.8) are both greater than 1/2, the set Z \ g n Z \ (g n ; I) has positive measure for all suciently large n. Remark. Our proof of Proposition 5.4 follows the outlines of Sullivan's proof in [Su82] (see also [Su81]) of the fact that the boundary action of a Kleinian group is of type III 1 on its conservative part. Ergodicity of the horocycle ow is just a reformulation of this statement (type III 1 actions are precisely those for which the Radon{Nikodym extension is ergodic). However, the application to horocycle ows is never explicitly mentioned in [Su82]. The author was told about Sullivan's result by Jon Aaranson and Benji Weiss after his talk at the Hebrew University in March 1998 where another approach to proving ergodicity of the horocycle ow was presented (see below Section 7). 6. Boundary actions and harmonic functions Given a Fuchsian group G, denote by? =?(G) the space of ergodic components of the G-action 2 with respect to the smooth ( visibility) measure type [m], i.e., with respect to the common type of the measures m x. In the notations below we shall usually overline all the objects connected with the space of ergodic components. By denition, there exists a measurable projection p 2!?; 7! and a measure type [m] = p([m]) on? such that (6.1) p(g) = p() 8 g 2 G; [m]-a.e. g 2 and (6.2) The linear operator F 7! F p is an isometry between L 1 (?; [m]) and the subspace of G-invariant functions in L 1 (@H 2 ; [m]).

17 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 17 The subsets p?1 2 ; 2? are called the ergodic components. We emphasize that the projection p and the space? itself are dened in the measure category only, i.e., up to measure 0 subsets. Put m x = p(m x ). Then for any x 2 H 2 there exists a mod 0 unique family of conditional probability measures x; 2? concentrated on the ergodic components p?1 () such that (6.3) m x = Z x dm x () ; i.e., hf; m x i = Z hf; x i dm x() 8 F 2 L 1 (@H 2 ; [m]) : In particular, if F 2 L 1 (?; []), and F is the corresponding G-invariant function 2 (6.2), then (6.4) hf ; m x i = hf; m x i 8 x 2 H 2 : Note that by (6.1) m gx = p(m gx ) = p(gm x ) = p(m x ) = m x for any x 2 H 2 and g 2 G, so that we can consider the measures m x = m x as associated with the points x = Gx of the quotient surface M = GnH 2. The space of the ergodic components? and decomposition (6.3) also admit an interpretation in terms of harmonic functions on H 2 which we are going to discuss now. Given a complete Riemannian manifold N, denote the space of extreme rays of the convex cone H + (N) of positive harmonic functions on N. For a point let be the corresponding (multiplicative) harmonic cocycle on N (see Remark 2 at the end of Section 2), i.e., (x; y) = '(y)='(x) for any function ' from the ray. Fixing a reference point o 2 N allows one to choose from any ray the minimal harmonic function ' o(x) = (o; x) normalized by the condition ' o(o) = 1. Any function f 2 H + (N) admits then a unique decomposition (6.5) f(x) = ' o(x) d f o () = (o; x) d f o () ; where o f is a nite positive measure so that (6.5) is a convex isomorphism between H + (N) and the cone of nite positive measures However, this isomorphism depends on the choice of a reference point o (i.e., of the choice of the convex base ff : f(o) = 1g of the cone H + (N)). For another point o 0 2 N f(x) = Z ' o(x) d f o () = Z (o; x) d f o () = Z (o 0 ; x) (o; o 0 ) d f o () ;

18 18 VADIM A. KAIMANOVICH whence by the uniqueness of the decomposition (6.5) (6.6) d f o 0 () = (o; o 0 ) = ' o(o 0 ) : d f o The probability measure o = 1 o representing the constant harmonic function 1(x) 1, is called the harmonic measure with the pole at the point o. All measures x ; x 2 N are pairwise equivalent, and by (6.6) the multiplicative Radon{Nikodym cocycle of the family of harmonic measures coincides with the harmonic cocycle. The common type [] of the measures x is called the harmonic measure type Denote by H 1 (N) the space of bounded harmonic functions on N with the sup-norm. For any f 2 H 1 (N) there exists a unique decomposition (6.5) with the representing measure (which is no longer necessarily positive) f o sandwiched between?kfk o and kfk o. In particular, the measure f o is absolutely continuous with respect to o. As it follows from (6.6), the Radon{Nikodym derivative (6.7) b f() = d f o d o () does not depend on the choice of o, so that for bounded harmonic functions the representation (6.5) takes the form (6.8) f(x) = Z (o; x) d f o () = Z dx d o () d f o () = h b f; x i : Formulas (6.7) and (6.8) (the latter one is called the Poisson formula) establish an isometry between the space H 1 (N) and the space L 1 (@N; []). The considered as a measure space with the family of harmonic measures x on it is called the Poisson boundary of the manifold N. The group of isometries Iso(N) acts by automorphisms on the cone H + (N). Since the family of harmonic cocycles is obviously Iso(N)-invariant, for any g 2 Iso(N) and f 2 H + (N) gf go = g f o : In particular, if f is bounded, then by (6.7) d (gf) = dgf o d o = dgf go d go = dgf o dg o = g b f ; so that the isometry between the spaces H 1 (N) and L 1 (@N; []) established by the Poisson formula (6.8) is Iso(N)-equivariant. For the hyperbolic plane H 2 the space of extreme harmonic rays is naturally identied with the boundary circle (which is why we use the same 2 in both cases), the harmonic cocycle is (6.9) (o; x) = e (x;o) ;

19 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 19 the harmonic measures 2 are the visibility measures m x, and the Poisson formula (6.5) coincides with the classical Poisson formula for bounded harmonic functions in the unit disk (see Remark 2 at the end of Section 2), For the quotient surface M = GnH 2 there is an obvious one-to-one correspondence between harmonic functions on M and G-invariant harmonic functions on H 2 consisting in lifting from M to H 2, so that we can consider the cone H + (M) (resp., the space H 1 (M)) as a subcone (resp., subspace) of the cone H + (H 2 ) (resp., of the space H 1 (H 2 )) consisting of all G-invariant functions. Combining formulas (6.4) and (6.8), we obtain (cf. [Ka95]) Proposition 6.1. The Poisson boundary of the quotient surface M = GnH 2 is isomorphic to the space? of ergodic components of the G-action 2 endowed with the family of harmonic measures m x ; x 2 M. Corollary. The action of the group G 2 is ergodic if and only if the quotient surface M = GnH 2 has no non-constant bounded harmonic functions. The denition of the conditional measures o and formula (6.6) imply that for any o; x 2 H 2 and [m]-a.e. 2 G (o; x) = dm x dm o () = 2 dm x dm o () d o = 2 (o; x) d o ; where ; are the harmonic cocycles on M (we use the same notation for their lifts to H 2 ). Therefore, we obtain Proposition 6.2. For any xed point o 2 H 2 the conditional measures o ; 2? are the representing measures of the G-invariant harmonic functions ' o(x) = (o; x) in their decomposition (6.5) into an integral of minimal harmonic functions ' o(x) = (o; x) on H 2. Remark. Another proof of Proposition 6.2 consists in the following observation: in order to decompose 1 2 one can rst decompose 1 in the smaller cone of G-invariant harmonic functions (i.e., and then further decompose the minimal G-invariant harmonic functions on H 2 2. Proposition 6.3. The dissipative part 2 of the action of a Fuchsian group G on the boundary 2 is the set (6.10) 8 < : : X g2g 9 = e (go;o) < 1 ; : Proof. Since the action of G 2 is free mod 0, by denition D is the set of all points 2 such that the conditional measure o; = p() is concentrated on the

20 20 VADIM A. KAIMANOVICH orbit G. Therefore, by Proposition 6.2, 2 D i a countable sum of the functions ' g o ; g 2 G is a G-invariant function (which is then necessarily minimal in the cone of positive G-invariant harmonic functions). It remains to check when a G-invariant harmonic function f on H 2 can be presented as a weighted sum of minimal harmonic functions ' o along the G-orbit of a certain boundary point g 2, i.e., when (6.11) f(x) = X g2g g g (o; x) 8 x 2 H 2 ; where g are certain positive coecients. Uniqueness of the decomposition (6.11) and G-invariance of the function f imply that the coecients g must have the form g = C g (go; o) = C (o; g?1 o) for a certain constant C > 0. Then niteness of the function f, i.e., summability of the family f g g, is by (6.9) equivalent to convergence of the series (6.10). Remark. This result was rst proved in [Po76, Theorem 1] (see also the discussion in [Po82, Section 1]). The proof in [Po76] uses analytic properties of Blaschke products and does not immediately carry over to the higher dimensional situation. On the contrary, our proof uses just the coincidence of the Busemann and the harmonic cocycles, and therefore is verbatim valid for the action of an arbitrary discrete subgroup of Iso(H d+1 ) on the boundary sphere S d (endowed with the Lebesgue measure type). Combining the results of this Section with Lemma 3.2 and Theorem 4.2 we obtain Theorem 6.4. For a Fuchsian group G the following properties are equivalent: (i) The horocycle ow on the quotient surface M = GnH 2 is ergodic with respect to the Liouville invariant measure; (ii) The action of G on the boundary 2 is ergodic with respect to the smooth measure type; (iii) The linear action of G on R 2 : is ergodic; (iv) There are no non-constant bounded harmonic functions on the quotient surface M. Theorem 6.5. Given a Fuchsian group G, (i) The horocycle ow on the quotient surface M = GnH 2 is conservative; (ii) The action of G on the boundary 2 is conservative with respect to the smooth measure type [m]; (iii) P g2g e (go;o) = 1 for [m]-a.e. 2 and a certain ( any) reference point o 2 H 2. Theorems 6.4 and 6.5 completely solve the problem of relationships between the ergodic properties of the horocycle ow and the ergodic properties of the corresponding boundary action. For the geodesic ow on surfaces of constant negative curvature the answer to these questions is contained in the following

21 ERGODIC PROPERTIES OF THE HOROCYCLE FLOW 21 Hopf{Tsuji{Sullivan Theorem [Su81]. Given a Fuchsian group G, the following properties are equivalent (i) The geodesic ow on M is ergodic with respect to the Liouville invariant measure; (ii) The geodesic ow on M is conservative with respect to the Liouville invariant measure; (iii) The Brownian motion on M is recurrent; (iv) The Poincare series P g2g ed(o;go) diverges for a certain ( any) reference point o 2 H 2 ; (v) The action of G 2 H 2 2 n diag is ergodic with respect to the smooth measure type [m] [m]; (vi) The action of G 2 H 2 2 n diag is conservative with respect to the smooth measure type [m] [m]. In the case of the horocycle ow (unlike for the geodesic ow) ergodicity and conservativity are not equivalent (for examples see Section 8 below). Apart of that Theorems 6.4 and 6.5 are almost parallel to the Hopf{Tsuji{Sullivan Theorem (although the underlying ideas are dierent). In particular, ergodicity admits an interpretation in terms of the basic properties of the Brownian motion for both \geometric" ows on M: the geodesic ow is ergodic i the Brownian motion on M is recurrent, whereas the horocycle ow is ergodic i the Brownian motion on M has no bounded harmonic functions (i.e., the sample paths have no non-trivial stochastically signicant behaviour at innity). Another similarity is between conditions (ii) and (iv) of the Hopf-Tsuji-Sullivan theorem (conservativity of the geodesic ow is equivalent to divergence of the Poincare series) and conditions (i) and (iii) of Theorem 6.5 (conservativity of the horocycle ow is equivalent to a.e. divergence of the \Poincare{Busemann" series (6.10)). 7. Covering surfaces In this Section we consider the situation when there are two Fuchsian groups G and G such that G is a normal subgroup of G. In terms of the associated quotient surfaces M = GnH 2 and M = GnH 2 it means that we have the projections H 2! M! M ; where M is a regular cover of M with the deck group T = G=G. If M is \small", so that the geometry of the surface M is close to that of the group T, one might expect to connect the ergodic properties of the boundary action of G with the algebraic properties of the deck group T. The strongest condition on M is its compactness, then goes niteness of the volume of M, and the weakest condition on M which we consider here is the ergodicity of the geodesic ow on M, or, equivalently, the recurrence of the Brownian motion on M (see the Hopf{Tsuji{Sullivan Theorem in Section 6 for other equivalent reformulation). In this case we call the surface M recurrent and the corresponding Fuchsian group G corecurrent. Note that recurrence of M implies absence of non-constant bounded harmonic functions on M, i.e., ergodicity of the action of G 2 (see Proposition 6.1).