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ISOLATIONS OF GEODESIC PLANES IN HYPERBOLIC

3-MANIFOLDS

AMIR MOHAMMADI AND HEE OH

Abstract. We present a quantitative isolation property of properlyimmersed geodesic planes in a geometrically finite hyperbolic 3-manifold.Our estimates are polynomial in the shadow constants, tight areas, anddensities of the Bowen-Margulis-Sullivan measures of geodesic planes.The proof is based on the quantitative non-concentration property ofthe Patterson-Sullivan density and the analysis of a Margulis functionmeasuring the distance between two planes.

Contents

1. Introduction 12. Notation and preliminaries 73. Shadow constants 84. Tight area of a geodesic surface 155. Averaging operator 176. A linear algebra lemma 217. Height function ω 228. Averaging operator for the height function 259. Construction of a Margulis function 2810. Quantitative isolation of a closed orbit 3411. Appendix: Proof of Theorem 1.1 in the compact case 35References 37

1. Introduction

Let H3 denote the hyperbolic 3-space, and letG = PSL2(C) ' Isom+(H3).Any oriented complete hyperbolic 3-manifold can be presented as the quo-tient M = Γ\H3 where Γ is a torsion-free discrete subgroup of G. A geodesicplane in M is the image of a totally geodesic immersion H2 → M . LetH := PSL2(R). Via the identification of X := Γ\G with the oriented framebundle FM , a geodesic plane is the image of a unique PSL(2,R)-orbit underthe base point projection map

π : X ' FM →M.

The authors were supported in part by NSF Grants.

1

2 AMIR MOHAMMADI AND HEE OH

Moreover a properly immersed geodesic plane in M corresponds to a closedH-orbit in X.

In this paper, we obtain a quantitative isolation result of a closed H-orbit for geometrically finite hyperbolic 3-manifolds. We fix a left invariantRiemannian metric on G, which projects to the hyperbolic metric on H3.This induces the distance d on X so that the canonical projection G → Xis a local isometry. We use this Riemannian structure on G to define thevolume of a closed H-orbit in X. For a closed subset S ⊂ Γ\G and ε > 0,B(S, ε) denotes the ε-neighborhood of S.

Finite volume case. We first state the result for the finite volume case.In this case, Ratner [15] and Shah [18] independently showed that every H-orbit is either closed or dense in X. Moreover, every closed orbit has finitevolume and there are only countably many closed H-orbits in X. Mozesand Shah [13] proved that an infinite sequence of closed H-orbits becomesequidistributed in X. Our questions concerns the following quantitativeisolation property: for a given closed H-orbit Y in X,

(1) how close can Y approach another closed H-orbit?(2) given ε > 0, what portion of Y enters into in the ε-neighborhood of

another closed H-orbit?

When X is not compact, the distance between two closed H-orbits maybe zero, e.g., if they both have cusps going into the same cuspidal end of X.To remedy this issue, we use the thick-thin decomposition of X.

For ε > 0, we denote by Xε the ε-thick part of X, i.e.,

Xε = x ∈ X : inj(x) ≥ ε

where inj(x) denotes the injectivity radius of x.It turns out that, in the finite volume case, the volume of a closed orbit

is the only complexity which measures the quantitative isolation property.The following theorem was proved by Margulis in an unpublished note:1

Theorem 1.1 (Margulis). Let Γ be a lattice in G. Let Y 6= Z be closedH-orbits in X.

(1) For all 0 < ε < 1,

d(Y ∩Xε, Z) ε9 Vol(Y )−3 Vol(Z)−3.

(2) For any 0 < ε < 1,

mY (Y ∩B(Z, ε)) ε1/3 Vol(Z)

where mY denotes the probability H-invariant measure on Y .

In both statements, the implied constant depends only on Γ.

1The notation A B and A B means A ≤ CB and A ≥ CB for some absoluteconstant C > 0 depending only on Γ.

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Remark 1.2. By recent works ([12], [1]), only when Γ is an arithmeticlattice, there may be infinitely many closed H-orbits. Theorem 1.1 is alsoproved in [5, Lemma 10.3], based on the effective ergodic theorem whichrelies on the arithmeticity of Γ via uniform spectral gap on closed H-orbits.Margulis’s proof does not rely on the arithmeticity and yields a sharper ex-ponent. It is based on the construction of a certain function which measuresthe distance from a closed orbit, and also satisfies a coarse version of aninequality reminiscent to the one satisfied by super harmonic functions. Asimilar function appeared first in the work of Eskin, Mozes and Margulis inthe study of quantitative Oppenheim conjecture [7], and later in [6], [4] and[8].

Infinite volume case. We now turn to the case of a geometrically finitehyperbolic 3-manifold M = Γ\H3 of infinite volume. We denote by coreMthe convex core of M , and fix εX to be minq∈coreM inj(q) if coreM is com-pact, and the maximum ε0 so that the ε0-thin part of coreM consists ofdisjoint horoballs, otherwise. We call the εX -thick part of coreM the com-pact core of M . We denote by RFM the renormalized frame bundle and forε > 0, consider the following ε-thick part of RFM :

RFMε := RFM ∩Xε.

Let Y = yH be a closed H-orbit and let ∆Y := StabH(y). The asso-ciated hyperbolic surface SY := ∆Y \H2 is always geometrically finite [14].We assume SY is non-elementary; otherwise, we cannot expect an isolationphenomenon for Y , as there may be a continuous family of parallel closedelementary H-orbits.

Let 0 < δY ≤ 1 denote the critical exponent of SY . We define

(1.3) δY =

δY Y has no cusp

2δY − 1 otherwise;

note that 0 < δY ≤ 1, and δY = 1 if and only if SY has finite area.We now state the main result of this paper, which generalizes Theorem

1.1 to all geometrically finite manifolds:

Theorem 1.4. Let M be a geometrically finite hyperbolic 3-manifold. Let Ybe a non-elementary closed H-orbit in X, and denote by mY the probabilityBowen-Margulis-Sullivan measure on Y .

(1) There exists N > 0 such that for all 0 < ε < εX and for any non-elementary closed H-orbit Z 6= Y ,

(1.5) d(Y ∩ RFMε, Z)(

vY,ε(max1, sY ) · areat Z

)N/δYwhere• vY,ε = miny∈Y ∩RFMεmY (BY (y, ε)) where BY (y, ε) is the ε-ball

around y in the induced metric on Y .• sY is the shadow constant of SY ;


• areat Z is the tight area of SZ relative to M .(2) There exists N > 0 such that for all 0 < ε < εX ,

mY (Y ∩B(Z, ε)) (max1, sY )N · εδY /N · (areat Z)N

In both statements, the exponent N and the implied constants depend onlyon Γ.

We now introduce geometric invariants of closed orbits: the shadow con-stant and the tight area of a properly immersed geodesic surface. WhenX has finite volume, it turns out that the shadow constant sY is 1, andthe tight area of Z satisfies areat Z areaZ; hence Theorem 1.4 recoversTheorem 1.1. For a general geometrically finite case, these notions are new.

Definition 1.6 (Shadow constant). Let ΛY ⊂ ∂H2 denote the limit setof ∆Y , νp : p ∈ H2 denote the Patterson-Sullivan density for ∆Y , andBp(ξ, ε) denote the ε-neighborhood of ξ ∈ ∂H2 with respect to the Gromovmetric at p. We define the shadow constant of SY as follows:

(1.7) sY := supξ∈ΛY ,p∈[ΛY ,ξ],0≤ε≤1

νp(Bp(ξ, ε))

εδY νp(Bp(ξ, 1))

,

where [ΛY , ξ] is the union of all geodesics connecting points in ΛY to ξ.

Observe that the shrinking exponent of νp(Bp(ξ, ε)) is given by δY ratherthan by δY . We show that 0 < sY < ∞ in Corollary 3.14; moreover, notethat the above definition is an invariant of Y , i.e., it is independent of thechoice of y, see Lemma 3.16.

Definition 1.8 (Tight area of S). For a properly immersed geodesic surfaceS of M , we define the tight-area of S relative to M as follows:

areat(S) := area(S ∩N (coreM))

where N (coreM) := p ∈ M : d(p, q) ≤ inj(q) for some q ∈ coreM is thetight neighborhood of coreM .

Note that the tight area essentially amounts to the area of the portion ofS which enters either the cuspidal regions of coreM , or the 1-neighborhoodof the compact core of M . For M geometrically finite, the flares of S canenter the cuspidal regions of coreM essentially only in cusp-like shapes, andthis geometric feature implies the finiteness of the tight area of S relative toM (Lemma 4.6).

Remark 1.9. (1) If Y is convex cocompact, then vY,ε ε1+2δY . Ingeneral,

ε1+2δY vY,ε ε4δY −1

where the implied constants depend only on Y (cf. [9]). For instance,if Y has finite volume, vY,ε ε3 Vol(Y )−1 with the implied constantindependent of Y .

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(2) Our proof yields a more general version of Theorem 1.4(1): Lety ∈ Y0 ∩Xε and z ∈ Z. If y 6∈ BZ(z, ε), then

d(y, z)(

mY (BY (y, ε))

(max1, sY ) · areat Z

)N/δYwhere Z is allowed to be equal to Y .

A geometrically finite hyperbolic 3-manifold M is called acylindrical ifits compact core does not have any essential discs or cylinders which arenot boundary parallel. Note that this is a topological condition. When Mis convex cocompact acylindrical, there is a uniform positive lower boundfor δY = δY for all non-elementary closed H-orbits Y [10]; therefore thedependence of δY can be removed in Theorem 1.4.

Examples of X with infinitely many closed H-orbits are provided by thefollowing theorem which can be deduced from ([10], [11], [3]):

Theorem 1.10. Let M0 be an arithmetic hyperbolic 3-manifold with a prop-erly immersed geodesic plane. Any geometrically finite acylindrical hyper-bolic 3-manifold M which covers M0 contains infinitely many non-elementaryproperly immersed geodesic planes.

It is easy to construct examples of M satisfying the hypothesis of thistheorem. For instance, if M0 is an arithmetic hyperbolic 3-manifold witha properly imbedded compact geodesic plane P , M0 is covered by a geo-metrically finite acylindrical manifold M whose convex core has boundaryisometric to P .

We discuss some of the main ingredients of the proof of Theorem 1.4.First consider the case when X = Γ\G is compact. Let ε0 = εX/2. Let Yand Z be two closed orbits of H in X. Fixing y ∈ Y , the set

IZ(y) := v ∈ isl2(R)− 0 : ‖v‖ ≤ ε0, y exp(v) ∈ Z

keeps track of all points of Z ∩B(y, ε0) in the transversal direction to H.The following function fZ : Y → [2,∞), called the Margulis function,

encodes the information on the distance d(y, Z):

fZ(y) =

∑v∈IZ(y) ‖v‖−1/3 if IZ(y) 6= ∅

ε−1/30 otherwise

.

Note that even though we are eventually interested in controlling the size ofthe smallest v ∈ IZ(y), the above definition of fZ as a sum is a more stableobject to consider. Indeed, for h ∈ H, the value fZ(yh) can be traced backfrom the value fZ(y) and the size of h, whereas the smallest vector in IZ(yh)and the smallest vector in IZ(y) are related only via an a priori unknownconstant depending on Z; namely, we would need to choose ε0 = ε0(Z) sothat the ε0-neighborhood of y contains at most one sheet of Z.

Since fZ(y) and fZ(yh) are comparable for h ∈ H near identity, Theorem1.1 follows if we can show that the value of fZ is controlled by the appropriate


complexity of the orbit Z on average, i.e.,

(1.11) mY (fZ) Vol(Z)

where mY is the H-invariant probability measure on Y .The estimate (1.11) is proved by establishing the following super-harmonicity

type inequality: there exist t > 0 and b > 1 such that for all y ∈ Y ,

(1.12) AtfZ(y) ≤ 1

2fZ(y) + bVol(Z)

where (AtfZ)(y) =∫ 1

0 fZ(yusat)ds, us = ( 1 0s 1 ) and at =

(et/2 0

0 e−t/2

).

The proof of (1.12) is based on the simple linear algebra lemma, whichsays that in a finite dimensional irreducible representation ρ : SL2(R) →GL(V ) (we apply it to the adjoint representation), the norm ‖ρ(usat)(v)‖(for t 1) of a non-zero vector v ∈ V grows on average for s ∈ [0, 1]. Inthe Appendix §11 we give the proof of Theorem 1.1 in the compact case. Itmay be helpful to read the Appendix first.

When X is of finite volume with cusps, there is no lower bound for theinjectivity radius on X, and this forces to incorporate the height of y, whichis essentially ω(y) = inj(y)−1, in the definition of fZ , and hence establish(1.12) for a suitable power of ω as well.

Essentially, we would like to adapt this argument for a general geomet-rically finite manifold of infinite volume. As the H-invariant measure on Yis infinite in general, the correct measure to control the average behaviorof the orbit Y is the Bowen-Margulis-Sullivan probability measure, whichwe denote by mY . Recall that the complexity of a closed orbit Y was de-scribed only in terms of its volume in the finite volume case. In general,due to the fractal nature of the BMS measure on Y as well as the subtlefluctuation/dependence property of the Patterson-Sullivan density on theviewpoints, it turns out that there are more geometric invariants of Y whichplay roles in the quantitative isolation of Y relative to Z, such as the shadowconstant of Y and the tight area of Z. We remark that in most of the exist-ing literature, the operator At is defined as the averaging operator over largespheres in H2, but we consider averages over expanding pieces of the horo-cylces; horocyclic averages are more amenable to the change of variables anditeration arguments for Patterson-Sullivan measures. Finally, let us high-light that when ∆Y is a lattice, mY is H-invariant and (1.11) follows bysimply integrating (1.12) with respect to mY . In general, the BMS measureis not H-invariant, and we appeal to equidistribution results of expandinghorocycles, in order to conclude an estimate similar to (1.11) from inequali-ties like (1.12) which we show hold for all large t. Note, however, that sinceour function fZ is not a bounded function in the presence of cusps, we needto control the growth rate of fZ and ω to apply equidistribution theorems.

We end this introduction with an outline of the paper. In §2 we fix somenotation and conventions to be used throughout the paper. In §3 we discuss

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Sullivan’s shadow lemma with precise description of the multiplicative con-stants and show the finiteness of the shadow constant. In §4, we show thefiniteness of the tight area of a properly immersed geodesic surface. In §5,the definition of the averaging operator and a basic property of this operatorare discussed. In §6 we prove a lemma from linear algebra; this lemma is akey ingredient to prove a local version of our main inequality. §7 is devotedto the study of the height function in X. In §8, we discuss the averagingoperator applied to the height function. The definition of the desired Mar-gulis function is given in §9, where we also prove all the necessary estimatesfor the proof of Theorem 1.4. In §10, we give a proof of Theorem 1.4. Inthe Appendix §11, we provide a short proof of Theorem 1.1 in the compactcase.

2. Notation and preliminaries

We set G = PSL2(C) ' Isom+(H3), and H = PSL2(R). We fix H2 ⊂ H3

so that g ∈ G : g(H2) = H2 = H. We fix a point o∗ ∈ H2 ⊂ H3 and aunit tangent vector vo∗ ∈ To∗(H3) once and for all. Denote by K0 and M0

respectively the stabilizer subgroups of o∗ and vo∗ . The isometric action ofG on H3 induces identifications G/K0 = H3, G/M0 = T1 H3, and G = FH3

where T1 H3 and FH3 denote, respectively, the unit tangent bundle and theoriented frame bundle over H3.

Let A denote the following one-parameter subgroup:

A =

at =

(et/2 0

0 e−t/2

): t ∈ R

.

The right translation action of A on G induces the geodesic/frame flow onT1 H3 and FH3, respectively. Let v±o∗ ∈ S2 denote the forward and backwardend points of the geodesic given by vo∗ . For g ∈ G, we define

g± = g(v±o∗) ∈ S2.

Let Γ < G be a discrete subgroup. We denote by Λ the limit set of Γ.A point ξ ∈ Λ is called parabolic if ξ is fixed by a parabolic element of Γ,and radial iff there exists a geodesic ray ξt toward ξ which accumulates ona compact subset of M .

We assume that Γ is geometrically finite, which is equivalent to the condi-tion that Λ is the union of the radial limit points and parabolic limit pointsΛrad ∪ Λp.

We set M := Γ\H3 to be the associated hyperbolic 3-manifold, and set

X := Γ\G.

We denote by X0 = RFM the renormalized frame bundle, i.e.,

RFM = [g] ∈ X : g± ∈ Λ.


Thick-thin decomposition of X0. We fix a Riemannian metric d on Gwhich induces the hyperbolic metric on H3. By abuse of notation, we use dto denote the distance function on X induced by d, as well as on M . Fora subset S ⊂ ♠ and ε > 0, B♠(S, ε) denotes the set x ∈ ♠ : d(x, S) ≤ ε.When S = e, we simply write B♠(ε) instead of B♠(S, ε). When there isno room for confusion for the ambient space ♠, we omit the subscript ♠.

For x ∈ X, the injectivity radius of x, which we denote by inj(x), is thesupremum r > 0 such that the map g 7→ xg is injective on the ball BG(e, r).For ε > 0, we set

Xε := x ∈ RFM : inj(x) ≥ ε.As M is geometrically finite, either X0 is compact or X0−Xε is contained

in finitely many disjoint horoballs for all small ε > 0. Set(2.1)

ε0 =

minq∈RFM inj(q) if X0 is compact

maxε0 : X0 −Xε0 is contained in horoballs otherwise

Let ξ1, · · · , ξ` be representatives of Γ-orbits in Λp so that RFM − Xε0

is contained in a disjoint union of finitely many horoballs hξ1 , · · · , hξ` . Wedenote by N the expanding horospherical subgroup for A:

N =

us =

(1 0s 1

): s ∈ C

.

Each horoball hi = h(ξi) is of the form

hi = [gi]NA(−∞,−Ti]K0

for some Ti ≥ 1, where gi ∈ G is an element with g−i = ξi and A[−T,−∞) =at : −∞ < t ≤ −T.

Convention. We will use the notation A B when the ratio between thetwo lies in [C−1, C] for some absolute constant C ≥ 1. By an absoluteconstant, we mean C depends at most on G and Γ in general. We writeA B? (resp. A ?B) to mean that A ≤ CBL (resp. A ≤ C ·B) for someabsolute constants C > 0 and L > 0 —when Γ is a lattice, the exponent Ldepends only on G.

3. Shadow constants

Let H denote the collection of all closed H-orbits yH in X such thatStabH(y) is non-elementary, or equivalently, StabH(y) is Zariski dense in H.Fixing Y ∈ H, we will introduce various measures and constants associatedto Y in this section.

We choose, once and for all,

(3.1) y0 ∈ Y ∩Xε0 and ∆Y := StabH(y0).

As Γ is geometrically finite, the subgroup ∆Y is a geometrically finitesubgroup of H. We denote by ΛY ⊂ ∂H2 the limit set of ∆Y . We denote by

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0 < δY ≤ 1 the critical exponent of ∆Y , equivalently the Hausdorff dimen-sion of ΛY . We remark that δY = 1 if and only if Y has finite Riemannianvolume.

Let SY := ∆Y \H2. The convex core of SY , denoted by core(SY ), is givenby ∆Y \ hull(ΛY ). The compact core of SY is the minimal connected surfaceCY such that core(SY )− CY is a union of disjoint cusps. We denote by dYthe diameter of the compact core of SY .

Consider the proper map

fY : ∆Y \H = T1(SY )→ Y ⊂ X given by [h]→ y0h,

and let fY : SY →M be the induced proper immersion.Letting FY ⊂ H2 be the Dirichlet domain for ∆Y containing o∗, we fix

oY ∈ hull(ΛY ) ∩ FYso that its image [oY ] in SY is the base point of y0 via fY . As y0 ∈ Y ∩Xε0 ,[oY ] ∈ CY . Therefore, the choice of oY (equivalently the choice of y0) isunique up to the distance dY . If there is no confusion, we drop the index Yfrom the notation when referring to oY .

Patterson-Sullivan measure. We denote by νp = νY,p : p ∈ H2 thePatterson-Sullivan density for ∆Y , normalized so that νoY has mass one: forall γ ∈ ∆Y , p, q ∈ H2, ξ ∈ ΛY ,

dγ∗νpdνp

(ξ) = e−δY βξ(γ−1(p),p) and

dνqdνp

(ξ) = e−δY βξ(q,p)

where βξ(·, ·) denotes the Busemann function.Set

U :=

us =

(1 0s 1

): s ∈ R

= N ∩H

which is the expanding horocylic subgroup of H. Using the parametrizations 7→ us we may identify U with R. Note that for all s, t ∈ R,

a−tusat = uets.

For any h ∈ H, the restriction of the visual map g 7→ g+ is a diffeomor-phism between hU and S1−h−. Using this diffeomorphism, we can definea measure on hU :

(3.2) dµPShU (hur) = e

δY β(hur)+ (p,hur(p))dνp(hur)+;

this is independent of the choice of p ∈ H2.We simply denote this by

dµPSh (r) = dµPS

hU (hur).

Note that these measures depend on the U orbits but not on the individualpoints. By the ∆Y -invariance and the conformal property of the PS-density,we have

dµPSh (O) = dµPS

γh(O)


for any γ ∈ ∆Y and for any bounded Borel set O ⊂ R; therefore µPSy (O) is

well-defined for y ∈ ∆Y \H ' Y .For any y ∈ ∆Y \H and any s ∈ R, we have:

(3.3) µPSy ([−es, es]) = eδY sµPS

ya−s([−1, 1]).

Bowen-Margulis-Sullivan measure mY . We denote by mY the Bowen-Margulis-Sullivan probability measure on ∆Y \H = T1(SY ), which is theunique probability measure of maximal entropy (that is δY ) for the geodesicflow. We set

Y0 = supp(mY ),

i.e., the non-wandering set of the geodesic flow. We note that Y0 = f−1Y (Y ∩

RFM). We will also use the same notation mY to denote the push-forwardof the measure to the H-orbit Y in what follows.

Shadow lemma. The convex core of SY is the union of the compact coreC0 := CY and the union of disjoint horoballs, say, C1, · · · , Cm.

We choose the lifts Ci ⊂ FY ∩ hull(ΛY ), 0 ≤ i ≤ m, so that ∆Y \∆Y Ci =

Ci. In particular, ∂C0 intersects Ci as an interval. Let ξi ∈ ΛY be the basepoint of the horoball Ci, i.e., ξi = ∂Ci ∩S1. Let Fi be a minimal interval forthe action of Stab∆Y

(ξi) in ΛY − ξi such that oY = o lies on a geodesicconnecting a point of Fi to ξi.

For p ∈ H2, let dp denote the Gromov distance on S1: for ξ 6= η ∈ S1,

dp(ξ, η) = e−(βξ(p,y)+βη(p,y))/2

where y is any point on the geodesic connecting ξ and η.We set

Bp(ξ, r) = η ∈ S1 : dp(η, ξ) ≤ r.For t ≥ 0, we denote by V (p, ξ, t) the set of all η ∈ S1 such that the

distance between p and the orthogonal projection of η onto the geodesic[p, ξ) is at least t. For some absolute constant c0 > 0, we have

V (p, ξ, t+ c0) ⊂ Bp(ξ, e−t) ⊂ V (p, ξ, t− c0) for all t ≥ c0.

The following is a uniform version of Sullivan’s shadow lemma [21]. Theproof of this proposition can be found in [20, Thm. 3.4]; since the dependenceon the multiplicative constant is important to us, we give a sketch of theproof while making the dependence of constants explicit.

Proposition 3.4. There exists a constant c e?dY such that for all ξ ∈ ΛY ,p ∈ C0 and for all t > 0,

c−1 · νp(Fξt)βY e−δY t+(1−δY )d(ξt,∆Y (p)) ≤ νp(V (p, ξ, t))

≤ c · νp(Fξt)e−δY t+(1−δY )d(ξt,∆Y (p))

where

• ξt is the unit speed geodesic ray [p, ξ) so that d(p, ξt) = t;• Fξt = S1 if ξt ∈ C0, and Fξt = Fξi if ξt ∈ Ci for 1 ≤ i ≤ m;

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• βY := infη∈ΛY ,p∈C0νp(Bp(η, e

−dY )).

Proof. As we allow the constant c e?dY and dY = diam C0, it suffices toprove the claim when p = o ∈ C0. We fix ξ ∈ ΛY and let ξt denote the unitspeed geodesic from o to ξ. By the δY -conformality of νp : p ∈ H2, wehave

νo(V (o, ξ, t)) e−δY tνξt(V (o, ξ, t))

with an absolute implied constant. Therefore it suffices to show

νξt(V (o, ξ, t)) e(1−δY )d(ξt,∆Y (o))

while making the dependence of the implied constant explicit.

Claim A. If ξt ∈ ∆Y C0, then

(3.5) e−δY dY · infη∈ΛY

νo(B(η, e−dY )) νξt(V (o, ξ, t)) eδY dY

where the implied constants are absolute.By the assumption, there exists γ ∈ ∆Y such that d(ξt, γo) ≤ dY . Hence

it follows from the conformality of νp

e−δY dY νξt(V (o, ξ, t)) ≤ νγo(V (o, ξ, t)) = νo(V (γ−1o, γ−1ξ, t))

≤ eδdY νξt(V (o, ξ, t)).

The first inequality shows the upper bound since |νo| = 1. The lowerbound follows from the second inequality; indeed

V (γ−1o, γ−1ξ, t) = V (γ−1ξt, γ−1ξ, 0)

and the latter contains Bo(γ−1ξ, e−dY ), since d(o, γ−1ξt) ≤ dY .

Claim B. Let ξ be a parabolic limit point in ΛY . Assume that for some i,ξt ∈ Ci for all large t.

We claim:

(3.6) νξt(V (o, ξ, t)) eδY dY νo(Fξ) · e(1−δY )(d(ξt,∆Y (o))+dY );

and

(3.7) νξt(S1 − V (o, ξ, t)) eδY dY νo(Fξ) · e(1−δY )(d(ξt,∆Y (o))+dY ).

Let si ≥ 0 be such that ξsi ∈ ∂Ci. Then for all t ≥ si,

|d(ξt,∆Y (o))− (t− si)| ≤ dY .

Hence for (3.6), it suffices to show

(3.8) νξt(V (o, ξ, t)) e(1−δY )(t−si)νo(Fξ).

Note that

νξt(V (o, ξ, t)) ∑

p∈Γξ,pFξ∩V (o,x,t)6=∅

νξt(pFξ)


where Γξ = Stab∆Y(ξ). Since pFξ ∩V (o, ξt, t) 6= ∅, then d(o, po) ≥ 2t−k for

some absolute k,and hence

νξt(V (o, ξ, t)) ∑

d(o,po)≥2t

νξt(pFξ).

Let F ∗ξ denote the image of Fξ on the horocycle based at ξ passing through

o via the visual map. We use the fact that if d(o, po) ≥ 2t, then for all η ∈ Fξ,

|βη(p−1ξt, ξt)− d(o, po) + 2t| diamF ∗ξ ≤ dY

(cf. proof of [20, Lemma 2.9]). Since νξt(pFξ) =∫Fξe−δY βpη(ξt,pξt))dνξt(η),

νξt(Fξ) e−δY tνo(Fξ), we deduce, with multiplicative constant eδY dY ,∑d(o,po)≥2t

νξt(pFξ) ∑

d(o,po)≥2t

e2δY t−δY d(o,po)νξt(Fξ)

νo(Fξ)eδY t∑

d(o,po)≥2t

e−δY d(o,po)

νo(F )e(1−δY )t

using an := #p ∈ Γξ : n < d(o, po) ≤ n + 1 en/2 in the last estimate.This proves (3.6). shadow

The estimate (3.8) follows similarly now using

νξt(S1 − V (o, ξ, t)) ∑

d(o,po)≤2t

νξt(pF ).

and∑[2t]

n=0 ane−δY n e(1−2δY )t.

The next two remaining cases are deduced from Clams A and B.

Claim C. When ξ be a parabolic limit point, (3.6) holds with multiplicativeconstant e?dY (see the proof of [20, Prop. 3.6]).

Claim D. If ξt ∈ ∆Y Ci for some i, then (3.6) holds with multiplicativeconstant e?dY . (see the proof of [20, Lemma 3.8]).

Proposition 3.9. Let Y ∈ H. There exists RY e?dY such that for ally ∈ Y0, we have

R−1Y βY e

(1−δY )d(CY ,π(y)) ≤ µPSy ([−1, 1]) ≤ RY e(1−δY )d(CY ,π(y))

where π denotes the base point projection Y → SY = ∆Y \H2.

Proof. We give an argument which is a slight modification of the proof of[19, Prop. 5.1]. Since the map y 7→ µPS

y [−1, 1] is continuous on Y0 and

[h] ∈ Y0 : h− is a radial limit point of ΛY is dense in Y0, it suffices toprove the claim for y = [h] assuming that h− is a radial limit point.

Recall that µPSy ([−1, 1]) = eδY tµPS

ya−t([−e−t, e−t]) for all t ∈ R. Let t ≥ 0

be the minimal number so that π(ya−t) ∈ CY . Then

(3.10) d(π(y), CY ) ≤ d(π(y), π(ya−t)) ≤ dY + d(π(y), CY ).

13

On the other hand,

µha−t [−e−t, e−t] νha−t(o)(V (ha−t(o), h+, t))

with an absolute implied constant (cf. [20, Lemma 4.4]).Since ya−t ∈ CY , Fha−t = S1. So νha−t(Fha−t) = |νha−t | 1. Therefore,

for some implied constant e?dY , we have

βY e−δY t+(1−δY )d(π(y),π(ya−t)) νha−t(o)(V (ha−t, h

+, t))

e−δY t+(1−δY )d(π(y),π(ya−t)).

This estimate and (3.10), therefore, imply that

βY e(1−δY )d(π(y),CY ) µPS

y ([−1, 1]) e(1−δY )d(π(y),CY )

with the implied constant e?dY , proving the claim.

The shadow constant sY . We set

(3.11) δY =

δY if Y is convex cocompact

2δY − 1 otherwise

Note that 0 < δY ≤ 1 since δY > 1/2 when Y is not convex cocompact.We use the following variant of Sullivan’s shadow lemma, obtained by

Schapira-Maucourant ([21], [19]):

Corollary 3.12. We have

0 < supy∈Y0,0≤ε≤1

µPSy ([−ε, ε])

εδY µPS

y ([−1, 1])≤ R2

Y · β−1Y <∞,

where RY is as in Proposition 3.9.

Proof. By (3.3), we have µPSy ([−ε, ε]) = εδY µPS

ya− log ε([−1, 1]). Hence the case

when Y is convex cocompact follows by Proposition 3.9.Now suppose Y has a cusp. Let y ∈ Y0; using the triangle inequality, we

get that d(ya− log ε, CY )− d(y, CY ) ≤ − log ε. Therefore, by Proposition 3.9we have

µPSya− log ε

([−1,1])

µPSy ([−1,1])

≤ R2Y β−1Y · e

(1−δY )(d(ya− log ε,CY )−d(y,CY ))

≤ R2Y · β−1

Y · εδY −1.

In consequence, we have

µPSy ([−ε,ε])µPSy ([−1,1])

≤ R2Y · β−1

Y · ε2δY −1

which establishes the claim.

Recall that the shadow constant of Y was defined in (1.6):

(3.13) sY = supξ∈ΛY ,p∈[ΛY ,ξ],0≤ε≤1

νp(Bp(ξ, ε))

εδY νp(Bp(ξ, 1))

.


Corollary 3.14. Set

(3.15) pY = max

1, supy∈Y0,0≤ε≤1

µPSy ([−ε, ε])

εδY µPS

y ([−1, 1])

∈ [1,∞].

Thenmax1, sY pY e?dY β−1

Y .

Proof. If y = [h], then µy([−r, r]) =∫e−δY βhu+r

(h,hur)dνh(o)(hu

+r ). Since

|βhu+r (h, hur)| ≤ d(e, ur),

we have that for all 0 < r < 1,

µy([−r, r]) νh(o)(Bh(o)(h+, r))

where the implied constant is independent of r. Hence pY max1, sY .The claimed inequality is proved in Corollary 3.12.

The constants dY and sY are defined using the group ∆Y , which in turnis defined using our fixed base point y0, see (3.1). We now show that thesetwo quantities are indeed independent of the choice of y0.

Lemma 3.16. Let y0 and ∆Y be as in (3.1). Let y = y0h−1 ∈ Y for some

h ∈ H and let ∆′Y = StabH(y). Define d′Y and s′Y as above using ∆′Y insteadof ∆Y . Then dY = d′Y and sY = s′Y .

Proof. Note that ∆′Y = h∆Y h−1. Therefore, hFY is a fundamental domain

for ∆′Y . Moreover, hC0 is compact, hCi is a horoball for every 1 ≤ i ≤ m,

and hCi ∩ hCj = ∅ for all 1 ≤ i 6= j ≤ m. In consequence, since the actionof h on H2 is an isometric action, we get that

d′Y = diam(hC0) = diam(C0) = dY .

We now show that s′Y = sY . Let νp : p ∈ H2 denote the Patterson-Sullivan density for ∆Y . Then the family ν ′p := h∗νh−1p is the Patterson-Sullivan density for ∆′Y . To see this, first note that the limit set of ∆′Yequals hΛY . Let now ξ ∈ ΛY , then

d(

(hγh−1)∗ν′p

)dν ′p

(hξ) =d(

(hγ)∗νh−1p

)dh∗νh−1p

(hξ)

=dγ∗νh−1p

dνh−1p(ξ) = e−δY βξ(γ

−1(h−1p),h−1p)

= e−δY βhξ(hγ−1h−1(p),p).

The above, in view of the uniqueness of the Patterson-Sullivan density (up toscalar), thus implies that ν ′p := h∗νh−1p is the Patterson-Sullivan densityfor ∆′Y .

Now for every 0 < ε ≤ 1 and every ξ ∈ ΛY , we have

ν ′p(Bp(hξ, ε)) = h∗νh−1p(Bp(hξ, ε)) = νh−1p(h−1Bp(hξ, ε)).

15

Moreover, by the equivariance properties of the Gromov distance, we haveh−1Bp(hξ, ε) = Bh−1p(ξ, ε). We conclude that sY = s′Y .

4. Tight area of a geodesic surface

For a closed subset Q ⊂M , we define the tight neighborhood of Q by

N (Q) := p ∈M : d(p, q) ≤ inj(q) for some q ∈ Q.For a properly immersed geodesic surface S of M , we define the tight-area

of S relative to M as follows:

areat(S) := area(S ∩N (coreM)).

Proposition 4.1. For a properly immersed non-elementary geodesic surfaceS of M , we have

areat(S) <∞.

Proof. Since S is a geometrically finite surface, S is the union of its convexcore and finitely many connected components, called flares. Since core(S)has finite area, it suffices to show that for the image of each flare, say F , ofS, the area of F ∩N (coreM) is finite.

Let R be the maximum of the injectivity radius of points in the compactcore of M . Since N (coreM) is contained in the R-neighborhood of the com-pact core of M and finitely many tight neighborhoods of disjoint horoballs,it suffices to show that the intersection of F with the tight neighborhood ofa horoball in coreM has finite area.

We denote by Mcpt ⊂ coreM the compact core of M Fix a horoball h incore(M), i.e., a connected component of core(M) −Mcpt. We may assume

that h is covered by the horoball based at ∞: h = (x1, x2, y) ∈ H3 : y > cfor some c > 0.

Since ∞ is a bounded parabolic fixed point, there exists b > 0 such that(x1, x2, y) : |x1| < b, |x2| < b, y > c projects onto h. Since inj(x) y−1 forx = (x1, x2, y) ∈ h, the tight neighborhood N (h) is covered by

hchimney := (x1, x2, y) : |x1| < b+ c0, |x2| < b+ c0, y > cfor some absolute constant c0 > 0.

Let p : H3 →M = Γ\H3 be the projection map. We set

hchimney := p(hchimney),

and call it the chimney in h. We claim that F ∩ N (h) is contained in theunion of an R0-neighborhood of core(S) and a cusp-like region with finitearea which is isometric to (x, y) ∈ H2 : |x| < b′, y > c and arises asF ∩ hchimney. This establishes the claim, since core(S) has finite area.

Fix a geodesic plane P ⊂ H3 which covers S and let ∆ = StabΓ(P ). Inthe rest of the proof, we denote by Λ ⊂ ∂P the limit set of ∆. Since S isproperly immersed, there exists some R1 > 0 so that

P ∩ p−1(S ∩B(Mcpt, 1)) ⊂ BP (hull(Λ), R1);


recall that S ∩B(Mcpt, 1) 6= ∅.We will show that there exists R2 > 0 depending only on R1 so that

(4.2) F ∩N (h) ⊂ BS(core(S), R2)⋃

hchimney.

Let z ∈ F ∩N (h) and let z ∈ P be any lift of z. Fix a complete geodesicξt ⊂ P with ξ0 = z. If any of the geodesic rays ξ(−∞,0) and ξ(0,∞) projectsinto h under p, then at least one of ξ±∞ belongs to the orbit Γ(∞). Asξ±∞ ⊂ ∂P , it follows that γ0(∞) ∈ ∂P for some γ0 ∈ Γ; hence γ−1

0 P is

a vertical plane. Since γh : γ ∈ Γ is a disjoint collection of horoballs and

γ−10 P ∩ h 6= ∅, we get

z ∈ p(γ−10 P ∩ hchimney) ⊂ hchimney.

Now suppose that neither of the geodesic rays ξ(−∞,0) and ξ(0,∞) is com-pletely contained in h under p. Since ∂h ⊂ Mcpt, there exists −∞ < t1 <0 < t2 < ∞ such that ξt1 , ξt2 ∈ P ∩ p−1(Mcpt). By the choice of R1, wehave ξt1 , ξt2 ∈ BP (hull(Λ), R1). For each i = 1, 2, let ξ′ti ∈ hull(Λ) be theunique closest points to ξti in hull(Λ). Let a be the union of three geodesicsegments:

a := (ξt1ξ′t1) ∪ (ξ′t1ξ

′t2) ∪ (ξ′t2ξt2).

Then a is a quasi-geodesic with both multiplicative and additive constantsdepending only on R1; note also that since hull(Λ) is convex, (ξ′t1ξ

′t2) ⊂

hull(Λ) and hence a ⊂ BP (hull(Λ), R1). Thus, by the Morse lemma fromhyperbolic geometry, there exists some R′1 > 0, depending only on R1,so that the Hausdorff distance between a and (ξt1ξt2) is at most R′1. Inparticular, we get that

z = ξ0 ∈ BP (hull(Λ), R1 +R′1).

By settingR2 := R1+R′1, we get z ∈ BS(core(S), R2), as desired in (4.2).

We record the following byproduct of the proof of the above proposition,which is of independent interest.

Corollary 4.3. For any properly immersed geodesic surface S ⊂ M , thereexists R ≥ 0 such that

S ∩N (core M) ⊂ BS(core(S), R)⋃N (h†chimney)

where h†chimney is the union of chimneys over all horoballs in core M .

This corollary says that the portion of S, especially of the flares of S,staying in the tight neighborhood of coreM can go to infinity only in cusp-like shapes, by visiting the chimneys of horoballs of coreM . This is nottrue any more if we replace the tight neighborhood of coreM by the 1-neighborhood of coreM . More precisely if Λ(Γ) contains a parabolic limitpoint of rank one which is not stabilized by any element of π1(S), then someinfinite area region of S can stay in the 1-neighborhood of coreM . This

17

situation may be compared to the presence of divergent geodesics in finitearea setting.

Tight volume of Z ∈ H. We consider the following neighborhood of X0:

(4.4) X ′0 := x ∈ X : ∃ x0 ∈ X0 with d(x, x0) ≤ inj(x0),and set

M ′0 := π(X ′0)

where π : X →M is the base-point projection map.Note that M ′0 ⊂ N (coreM), in particular, it is contained in the 1-

neighborhood of coreM , and hence has finite volume.We set

(4.5) τZ := max1, area(SZ ∩M ′0).Recalling the definition of the tight area of SZ from (1.8) and the fact

that M ′0 ⊂ N (coreM), note that

area(SZ ∩M ′0) ≤ areat(SZ).

This, together with the fact that areat SZ ≥ (ε3)2 where ε3 is the Margulisconstant for H3, implies that τZ areat(SZ).

Corollary 4.6. For Z ∈ H, we have

1 ≤ τZ areat(SZ) <∞.

5. Averaging operator

In this section we define an averaging operator and investigate some ofits basic properties. Fix a closed orbit Y ∈ H.

Definition 5.1. For each t ∈ R and ρ > 0, we define the operator At,ρ :C(Y0)→ C(Y0) by

(5.2) (At,ρψ)(y) :=1

µPSy ([−ρ, ρ])

∫ ρ

−ρψ(yurat)dµ

PSy (r).

For the sake of notational ease, we set At := At,1.

This is well-defined by the continuity of the map y 7→ µPSy ([−ρ, ρ]) for

ρ > 0.We remark that in [7], the averaging operator At was defined using the

integral over the translates SO(2)at, whereas we use the integral over thetranslates U[−ρ,ρ]at of a horocyclic piece. The proof of the following propo-sition, which is an analogue of [7, §5.3], is the main reason for our digressionfrom their definition, as the handling of the PS-measure on U is much hand-ier than that of the PS-measure on SO(2) in performing change of variables.

Proposition 5.3. For every σ > 1 there exists some 0 < c0 = c0(σ) < 1so that the following holds. Suppose that F : Y0 → [2,∞) is a continuousfunction satisfying the following properties:


(a) For all h ∈ BH(2) and y ∈ Y0, we have

σ−1F (y) ≤ F (yh) ≤ σF (y).

(b) There exists τ ≥ 2 and D0 = D0(τ) > 0 such that, for all y ∈ Y0

and 1 ≤ ρ ≤ 2, we have

Aτ,ρF (y) < c0 · F (y) +D0.

Then we have:

(1) For any 0 < c < 1, there exists t0 = t0(σ, c) > 0, and D = D(σ) > 0so that for all t > t0 and all y ∈ Y0, we have

AtF (y) < c · F (y) +D.

(2) For any y ∈ Y0, we have

lim supt→∞

AtF (y) ≤ 1 +D.

Proof. For any y ∈ Y0 and for any ρ > 0, set

by(ρ) := µPSy ([−ρ, ρ]).

For simplicity, we set by = by(1).In view of Proposition 3.9, we have

(5.4) supy∈Y0

by(2)

by≤ d · pY

where d is an absolute constant. Fix σ > 1, and let 0 < c0 = c0(σ) < 1 beso that

(5.5) κ := 2c0σdpY < 1.

Let D0 be as in the assumption (b) and put

(5.6) D := 2D0dpY .

In the sequel we will use the fact that∑

n≥1 e−nτ ≤ 1/2 — recall that

τ ≥ 2. The main step in the proof is the following estimate.

Claim 5.7. For any 1 ≤ ρ ≤ 2 and any y ∈ Y0, we have

(5.8) A(n+1)τ,ρF (y) ≤ κAnτ,ρ+e−nτF (y) + D.

Let us first assume Claim 5.7 and finish the proof. Iterating (5.8) withρ = 1 and using the fact that

∑e−nτ ≤ 1/2, we have that for any n ≥ 2,

AnτF (y) ≤ κn−1Aτ,1+∑e−jτF (y) + D(1 + κ+ · · ·+ κn−2)

≤ κn−1(c0F (y) +D0) + D(1 + κ+ · · ·+ κn−2)

≤ κnF (y) + D

where D = D1−κ and for the second inequality we used our assumption (b)

with ρ = 1 +∑e−jτ .

19

Repeatedly applying the assumption in part (a), there exists some σ1 =σ1(σ) > 1 so that

σ−11 F (y) ≤ F (yh) ≤ σ1F (y)

for all h ∈ ur : |r| ≤ 2a` : |`| ≤ τur : |r| ≤ 2 and all y ∈ Y0.Let nc = nc(σ) ≥ 1 be large enough so that κnσ1 < c for all n ≥ nc. Let

now t ≥ ncτ and write t = nτ + ` where 0 ≤ ` < τ . Then

AtF (y) =1

bz

∫ 1

−1F (yuranτ+`)dµ

PSy (r) ≤ σ1AnτF (y).

Since σ1κn < c, we get

AtF (y) ≤ σ1κnF (y) + σ1D ≤ cF (y) +D

where D = σ1D.In consequence, assuming Claim 5.7, we obtain part (1) with tc = ncτ

and D as above.To prove part (2), apply part (1) with c = 1/F (y).

Proof of Claim 5.7. We now turn to the proof of (5.8). To ease the notation,we prove this with ρ = 1; the proof in general is similar.

By our assumption in part (b), we have

(5.9) AτF (y) ≤ c0F (y) +D0 ≤(c0σ

by

∫ 1

−1F (yur)dµ

PSy (r)

)+D0.

Set ρn := e−nτ . Let [rj − ρn, rj + ρn] : j ∈ J be a covering of

[−1, 1] ∩ supp(µPSy )

with rj ∈ [−1, 1] ∩ supp(µPSy ) and with multiplicity bounded by 2. For each

j ∈ J , let zj := yurj . Then

(5.10)∑j

bzj (ρn) =∑j

µPSy (zju[rj−ρn,rj+ρn]) ≤ 2by(2).

Moreover, we can compute

A(n+1)τF (y) =1

by

∫ 1

−1F (yura(n+1)τ )dµPS

y (r)

≤ 1

by

∑j

∫ ρn

−ρnF (zjura(n+1)τ )dµPS

zj (r)

=1

by

∑j

∫ ρn

−ρnF (zjanτurenτaτ )dµPS

zj (r).(5.11)

We now make the change of variables t = renτ . Therefore, in view of (5.11),we have

A(n+1)τF (y) ≤ 1

by

∑j

bzj (ρn)

bzjanτ

∫ 1

−1F (zjanτutaτ )dµPS

zjanτ (t).


Applying (5.9) with the base point zjanτ , we get from the above that

(5.12) A(n+1)τF (y) ≤ 1

by

∑j

bzj (ρn)c0σ

bzjanτ

∫ 1

−1F (zjanτur)dµ

PSzjanτ (r)+

1

by

∑j

bzj (ρn)D0.

By (5.10), we have1

by

∑j

bzj (ρn)D0 ≤ D;

see (5.6). Therefore, reversing the change of variable, i.e., now letting t =e−nτr, we get from (5.12) the following:

A(n+1)τF (y) ≤ 1

by

∑j

c0σ

∫ ρn

−ρnF (zjutanτ )dµPS

zj (t) + D

≤ 2c0σ

by

∫ 1+ρn

(−1−ρn)F (yuranτ )dµPS

y (r) + D

=2c0σby(1 + ρn)

by · by(1 + ρn)

∫ 1+ρn

(−1−ρn)F (yuranτ )dµPS

y (r) + D

≤ κAnτ,1+ρnF (y) + D(5.13)

as κ ≥ supy∈Y02c0σby(2)

by.

Lemma 5.14. If F : Y0 → [2,∞) is a continuous function satisfying (1)and (2) in Proposition 5.3, then F ∈ L1(Y0,mY ).

Proof. The claim follows from the Birkhoff’s ergodic theorem as we nowexplicate. For every R > 2, let FR : Y0 → R be given by FR(y) :=minF (y), R. As FR is bounded, it belongs to L1(Y0,mY ). Let R0 belarge enough so that

∫FR dmY > 0 for all R > R0; fix some R > R0.

Therefore, in view of the A-ergodicity of mY , for mY -a.e. y ∈ Y0, we have

1T

∫ T

0FR(yat)dt→

∫FR dmY .

Hence, using Egorov’s theorem, for every ε > 0 there exists a subset Y ′ε ⊂ Y0

with mY (Y ′ε ) > 1 − ε2 and some Tε > 1 so that for every y ∈ Y ′ε and allT > Tε we have

1T

∫ T

0FR(yat)dt >

12

∫FR dmY .

Now by the maximal ergodic theorem [17, Thm. 17], if ε is small enough,there exists a subset Yε ⊂ Y ′ε with m(Yε) > 1 − ε so that for all y ∈ Yε wehave

µPSy r ∈ [−1, 1] : yur ∈ Y ′ε > 1

2µPSy ([−1, 1]).

21

Altogether, if y ∈ Yε and T > Tε, we have

(5.15)1

µPSy ([−1, 1])

∫ 1

−1

1

T

∫ T

0FR(yurat)dt dµ

PSy (r) >

1

4

∫FR dmY .

Fix some y ∈ Yε and apply Proposition 5.3(1) with this y and c = 1/4.Then for all large enough t, we have

1

µPSy ([−1, 1])

∫ 1

−1FR(yurat)dµ

PSy (r) ≤ 1

µPSy ([−1, 1])

∫ 1

−1F (yurat)dµ

PSy (r)

<1

4F (y) +D.

Averaging this over t and using (5.15), we get that∫FR dmY < F (y) + 4D.

Since R > R0 is arbitrary, the claim follows.

We will also use the following equidistribution theorem in the sequel.

Theorem 5.16 (Equidistribution of expanding horocycles [16], [14]). Forany bounded ψ ∈ C(Y0), y ∈ Y0, and ρ > 0, we have

limt→+∞

At,ρψ(y) = mY (ψ).

6. A linear algebra lemma

In this section, it is more convenient to use the isomorphism between Gand SO(Q) for the quadratic form

Q(x1, x2, x3, x4) = 2x1x4 + x22 + x2

3.

We consider the standard representation of G on R4. We have H =StabG(e3) ' SO(2, 1), and

A = at = diag(et, 1, 1, e−t) < H.

Set V := Re1 ⊕Re2 ⊕Re4. Then the action of H on V is isomorphic to theadjoint representation of H on its Lie algebra; in particular, it is irreducible.

Lemma 6.1. For any Y ∈ H, 0 < s <δY2 , and 1 ≤ ρ ≤ 2, we have

(6.2) supy∈Y0,‖v‖=1

1

µPSy ([−ρ, ρ])

∫ ρ

−ρ

1

‖vurat‖sdµPS

y (r) ≤ c1pY e−st/2

where c1 depends only on V .

Proof. Since µPSy [−ρ, ρ] = ρδY µPS

ya− log ρ[−1, 1] and Y0 is A-invariant, it suffices

to prove the claim for ρ = 1. We write

V = V − ⊕ V 0 ⊕ V +

where V 0 = v ∈ V : vat = v for all t ∈ R, V + denotes the sum ofeigenspaces with eigenvalue > 1 and V − denotes the sum of eigenspaceswith eigenvalue < 1 for A = at.


Let p+ : V → V + denote the natural projection. For any unit vectorv ∈ V and any ε > 0, define

D+(v, ε) = r ∈ [−1, 1] : |p+(vur)| ≤ ε.

We claim that the Lebesgue measure of D+(v, ε) is at most c · ε1/2 wherec is an absolute constant.

To see this, note that the map r 7→ p+(vur) is a polynomial map of degreeat most 2. Moreover, since ‖v‖ = 1, we have maximum of the absolute valueof the coefficients of p+ is bounded from below by an absolute constant. Inparticular, supr∈[−1,1] ‖p+(vur)‖ ≥ c′ for some absolute c′ > 0 (depending

only on 2). The claim now follows using Lagrange’s interpolation, see [2] fora more general statement.

Together with the fact that D+(v, ε) is a union of at most 2 intervals,this claim implies that D+(v, ε) may be covered by 1 many intervals of

length ε1/2. Now by the definition of pY in (3.15), we get

(6.3) µPSy (D+(v, ε)) ≤ c0pY ε

δY /2µPSy ([−1, 1])

where c0 is an absolute constant.

Recall that s < s0 :=δY2 . Note also that ‖vurat‖ ≥ et/2‖p+(vur)‖. In

consequence, for all 0 < ε ≤ 1/2 and all ‖v‖ = 1, we have

1

µPSy ([−1, 1])

∫r∈D+(v,ε)−D+(v,ε/2)

1

‖vurat‖sdµPS

y (r) ≤ c0pY εs0 · (et/2ε/2)−s

= c02spY εs0−se−st/2.

Summing up the geometric series we get

(6.4)1

µPSy ([−1, 1])

∫r∈D+(v,1/2)

1

‖vurat‖sdµPS

y (r) ≤ c0pY2−s − 2−s0

e−st/2.

On the other hand,

(6.5)1

µPSy ([−1, 1])

∫r∈[−1,1]−D+(v,1/2)

1

‖vurat‖sdµPS

y (r) ≤ 2e−st/2.

Combining (6.4) and (6.5) we get that

1

µPSy ([−1, 1])

∫ 1

−1

1

‖vurat‖sdµPS

y (r) ≤(

c0pY2−s−2−s0

+ 2)e−st/2,

as was claimed in the lemma.

7. Height function ω

In this section we use the thick-thin decomposition of RFM to define thenotion of height of x ∈ RFM . For this purpose, we continue to use theisomorphism between G and SO(Q), and consider the standard representa-tion of G on R4. By the choice of the quadratic form Q, Q(e1) = 0 and thestabilizer of e1 is equal to MN .

23

We assume that Γ is not convex cocompact, so that Λp 6= ∅; otherwise thecontent of this section is void. We fix ξ1, · · · , ξ` a set of Γ-representatives inΛp, and choose gi ∈ G so that g−i = ξi. Then

StabG(ξi) = giAM0Ng−1i .

For each 1 ≤ i ≤ `, let vi ∈ R4 be a unit vector on the line Re1g−1i . Then

(7.1) giM0Ng−1i = StabG(vi)

and by Witt’s theorem,

v ∈ R4 : v 6= 0,Q(v) = 0 = viG ' giM0Ng−1i \G.

Lemma 7.2. For each 1 ≤ i ≤ `, the orbit viΓ is a closed (and hencediscrete) subset of R4.

Proof. The condition ξi ∈ Λp implies that Γ\ΓgiM0N is a closed subset of

X. Equivalently, ΓgiM0N , as well as ΓgiM0Ng−1i is closed in G. Therefore,

its inverse giM0Ng−1i Γ is a closed subset of G. By (7.1), viΓ ⊂ R4 is a closed

subset of viG = v ∈ R4 : v 6= 0,Q(v) = 0.It is now enough to show that viΓ does not accumulate on 0. Suppose

on the contrary that there exists an infinite sequence viγ` → 0 for someγ` ∈ Γ. Using the Iwasawa decomposition G = giNAg

−1i K0, we may write

γ` = gin`at`g−1i k` with n` ∈ N, t` ∈ R and k` ∈ K0. Since

viγ` = et`(e1g−1i k`),

the assumption that viγ` → 0 implies that t` → −∞.On the other hand, as ξi ∈ Λp, StabΓ(ξi) = Γ ∩ giAM0Ng

−1i contains a

parabolic element, say, γ′. Note that n0 := g−1i γ′gi is then an element of N ,

as any parabolic element of AM0N belongs to N in the group G ' PSL2(C).Now observe that

γ−1` γ′γ` = k−1

` gia−t`(n−1` g−1

i γ′gin`)at`g−1i k` = k−1

` gi(a−t`n0at`)g−1i k`

as N is abelian.Since t` → −∞, the sequence a−t`n0at` converges to e. Since k−1

` gi is a

bounded sequence, it follows that, up to passing to a subsequence, γ−1` γ′γ`

is an infinite sequence converging to e, which contradicts the discreteness ofΓ.

We let ‖ · ‖ be the Euclidean norm on R4. For each 1 ≤ i ≤ `, define thefunction ωi : X0 → [2,∞) as follows: for [g] ∈ X0,

ωi([g]) = maxγ∈Γ

2,

1

‖viγg‖

;

this is well-defined by Lemma 7.2.

Definition 7.3 (Height function). Define the height function ω : X0 →[2,∞) by

ω(x) := max1≤i≤`

ωi(x).


It will be convenient to introduce the following notation:

Notation 7.4. Let Q ⊂ G be a compact subset.

(1) Let dQ ≥ 1 be the infimum of all d ≥ 1 such that for all g ∈ Q andv ∈ R4,

(7.5) d−1‖v‖ ≤ ‖vg‖ ≤ d‖v‖.

Note that dQ maxg∈Q ‖g‖, up to an absolute multiplicative con-stant.

(2) We also define cQ ≥ 1 to be the infimum of all c ≥ 1 such that forany x ∈ X0, g ∈ Q with xg ∈ X0, and for all 1 ≤ i ≤ `

(7.6) c−1ωi(x) ≤ ωi(xg) ≤ cωi(x).

We note that cQ maxg∈Q ‖g‖ up to an absolute multiplicativeconstant.

Proposition 7.7. For all x ∈ X0,

ω(x) inj(x)−1

with an absolute implied constant. In particular, ω : X0 → [2,∞) is a propermap satisfying that for all x ∈ RFM ,

ω(x) d(x,Xε0 ∩ RFM)

where the implied constant is independent of x ∈ RFM .

Proof. Recall that a horoball in X0 is of the form X0 ∩ [gj ]NA(−∞,T ]K0 forsome T 1. In view of the thick-thin decomposition, it suffices to show theclaim for [g] ∈ X0 ∩ [gj ]NA(−∞,T ]K0.

Let g = γggjua−tk where ua−tk ∈ NA(−∞,T ]K0 and γg ∈ Γ; so inj([g]) e−t. Note that

(7.8) e−t = ‖e1ua−tk‖ = ‖e1g−1j γ−1

g γggjua−tk‖ = ‖e1g−1j γ−1

g g‖.

In view of the definition of ω and ωi, the claim will follow if we show thefollowing: there exists some c 1 so that for all 1 ≤ i ≤ ` and γ ∈ Γ suchthat γ−1hi 6= γghj , we have

(7.9) ‖e1g−1i γg‖ ≥ c

where hj = gjNA(−∞,T ]K0.

Write g−1i = uiatiki and let τ ≥ 0 be so that |ti| < eτ for all i. Assume

now that there exists some i so that

‖e1g−1i γg‖ ≤ e−T−τ−1.

Then γg = giu′a−sg

−1i k′ where u′ ∈ N , k′ ∈ K0 where s ≥ T + τ + 1. In

view of the choice of τ , this implies that g ∈ γ−1hi. Since g ∈ γghj , we getthat i = j and γ−1hj = γghj .

This implies that (7.9) holds with c = e−T−τ−1 and finishes the proof.

25

Using [9, Prop. 5.5], one can show that the restriction of ωs to Y belongsto L1(Y,mY ) if and only if 0 < s < 2δY − 1. To keep this paper as self-contained as possible, however, we will use Lemma 5.14 for the estimates weneed in Proposition 8.6.

8. Averaging operator for the height function

In this section, we fix a closed H-orbit Y ∈ H and fix

0 < s <δY2.

The main goal of this section is to prove the following property of theaveraging operator applied to the height function ωs (restricted to Y0):

Theorem 8.1. (1) There exists 0 < D1 p?Y with the following prop-erty: for any 0 < c ≤ 1/2, there exists some 0 < t(c, pY ) | log c|+ log pY such that for any y ∈ Y0 and all t ≥ t(c, pY ),

(8.2) Atωs(y) ≤ c · ωs(y) +D1.

(2) For every y ∈ Y0, we have

(8.3) lim supt→∞

Atωs(y) ≤ 1 +D1.

The proof of this theorem is based on Proposition 5.3; indeed the functionωs satisfies the condition in Proposition 5.3(a) by (7.6). We will now showthat it also satisfies the condition in Proposition 5.3(b).

Lemma 8.4. For every 0 < c ≤ 1/2, there exist some 0 < τ | log c| +log pY and some 1 ≤ D2 p?Y /c

? so that for all y ∈ Y0 and all 1 ≤ ρ ≤ 2,

Aτ,ρωs(y) ≤ c · ωs(y) +D2.

Proof. Since Q(e1) = 0 and G = SO(Q), we have Q(e1g) = 0 for everyg ∈ G; in particular, there exists an absolute constant η > 0 so that forevery unit vector v ∈ e1G, we have

(8.5) ‖v − e3‖ ≥ η.

For every v ∈ R4, let us write v = v0 + v1 where v0 ∈ Re3 and v1 ∈ V =Re1 ⊕ Re2 ⊕ Re4.

Let v ∈ e1g be as above. Moreover, since R4 = Re3 ⊕ V as an H-representation and since e3 is H-invariant, we have vh = v0 +v1h ∈ Re3⊕Vfor all h ∈ H. Therefore,

1

µPSy ([−ρ, ρ])

∫ ρ

−ρ

1

‖vurat‖sdµPS

y (r) ≤ 1

µPSy ([−ρ, ρ])

∫ ρ

−ρ

1

‖v1urat‖sdµPS

y (r)

≤ c1pY e−st/2‖v1‖−s by Lemma 6.1

≤ d1pY e−st/2‖v‖−s by (8.5)

for some absolute constant d1 > 0.


Let τ be given by the equation:

d1pY e−sτ/2 = c.

We claim that the lemma holds with this choice of τ . To see this, we compareω(yuraτ ) and ω(y) for r ∈ [−2, 2]. Setting

Q := ur : |r| ≤ 2at : |t| ≤ τur : |r| ≤ 2,

we have cQ eτ p2Y /c

2, see (7.6).Recall the constant ε0 which satisfies

ε−10 sup

y∈Xε0∩Y0ω(y).

We consider two cases.Case 1: ω(y) ≤ 2cQ/ε0. In this case, for h ∈ Q with yh ∈ Y0,

ω(yh) ≤ 2c2Q/ε0.

Hence, the claim in this case follows if we choose D2 > 2c2Q/ε0.

Case 2: ω(y) > 2cQ/ε0. For some 1 ≤ i ≤ `, we have

ωi(y) > 2cQ/ε0, and hence y ∈ hi.

By the definition of cQ, we have

ωi(yh) > 2ε0, and hence yh ∈ hi

for all h ∈ Q with yh ∈ Y0. Choose g0 ∈ G so that y = [g0]. In view of (7.9)and (7.8), there exists γ ∈ Γ such that simultaneously for all h ∈ Q withyh ∈ Y0,

ω(yh) = ‖viγg0h‖−1.

Therefore, we have

Aτ,ρωs(y) =

1

µPSy ([−ρ, ρ])

∫ ρ

−ρ

1

‖viγuraτ‖sdµPS

y (r)

≤ c1pY e−sτ/2‖viγ‖−s = c · ωs(y).

The proof is now complete.

Proof of Theorem 8.1. The conditions in Proposition 5.3 holds with F = ωs;indeed, condition (a) follows from (7.6) and condition (b) holds in light ofLemma 8.4. The theorem thus follows from Proposition 5.3.

Equidistribution for an unbounded function ωs. Using Theorem 8.1we will deduce the following proposition from Theorem 5.16 and Lemma5.14.

Proposition 8.6. For any y ∈ Y0, we have

limt→∞

Atωs(y) = mY (ωs).

27

Proof. As was mentioned before, we will prove this proposition using theequidistribution Theorem 5.16. Note, however, that ω is not a boundedfunction and we may not apply the aforementioned equidistribution resultsdirectly. We use the fact that s < 2δY −1

2 and choose s0 so that s+s0 <2δY −1

2to approximate ωs by a bounded function.

Let us now turn to the details. Let R > 1 be a parameter. Let 0 ≤ χR ≤ 1be a continuous function which is equal to 1 on x ∈ RFM : ω(x) ≥ R+ 1and is equal to 0 on x ∈ RFM : ω(x) ≤ R. Set

hR := ωs − ωsχR;

the function hR is a continuous function with compact support.By Theorem 5.16, we have

(8.7) limtAthR(y) = mY (hR).

Moreover, in view of Theorem 8.1 that ωs satisfies (1) and (2) in Proposi-tion 5.3. Hence, using Lemma 5.14, we have

(8.8) |mY (ωs)−mY (hR)| → 0 as R→∞.

Let s1 = s + s0; then s1<2δY −1

2 . Note that, using ω−s0 ≤ R−s0 onω(x) ≥ R, we have

(8.9) (ωsχR)(y) ≤ R−s0 · ωs1(y),

and hence

At(ωsχR)(y) ≤ R−s0 ·Atωs1(y).

Now by Theorem 8.1(2), applied with s1, we have

lim supt

Atωs1(y) ≤ 1 +D1

which in view of the above computation implies

(8.10) At(ωsχR)(y) ≤ R−s0(1 +D1).

Therefore for any R > 1, using (8.7),

lim supt|Atωs(y)−mY (ωs)|

≤ lim supt|At(ωs · χR)|+ lim sup

t|At(hR)−mY (hR)|+ |mY (hR)−mY (ωs)|

≤ R−s0(1 +D1) + |mY (hR)−mY (ωs)| by (8.10).

Taking R→∞ and using (8.8), we get

lim supt|Atωs(y)−mY (ωs)| = 0.

The proof of complete.


Remark 8.11. Since no horoball can contain a complete geodesic, it followsthat Y0 intersects Xε0 . This fact, however, is not used in the discussion here.Indeed, similar to the proof of Theorem 1.4(2), we get a non-divergence resultfrom Theorem 8.1 and Proposition 8.6 with explicit constants depending onΓ and pY .

9. Construction of a Margulis function

Throughout this section, we fix Y, Z ∈ H and

0 < s <δY2.

The Lie algebra sl2(C) of G decomposes as sl2(R) ⊕ isl2(R). We writeV = isl2(R) and consider the action ofH on V via the adjoint representation;so vh = Ad(h−1)(v) for v ∈ V and h ∈ H. We use the relation g(exp v)h =gh exp(vh) which is valid for all g ∈ G, v ∈ V, h ∈ H.

For any E ≥ 1 and any y ∈ Y0, we define

(9.1) IZ(y,E) = v ∈ V : ‖v‖ ≤ ω(y)−Ep−EY , y exp(v) ∈ Z.

Recall the following notation from (4.4) and (4.5):

X ′0 = x ∈ X : ∃ x0 ∈ X0 with d(x, x0) ≤ inj(x), and

τZ = max1, area(SZ ∩ π(X ′0).

Lemma 9.2. There exist E,L, c0 ≥ 1 so that

supy∈Y0

#IZ(y,E) ≤ c0pLY · τZ .

Proof. Fix 0 < D1 p?Y as given by Theorem 8.1 for Y . Set

K := x ∈ Y0 : ω(x) ≤ 1 +D1.

Let E0 > 1 be given by the conditions that exp : sl2(C) → G is a localdiffeomorphism on the 2−E0-neighborhood of 0, and for any x ∈ X

infinj(x exp v) : ‖v‖ ≤ 2−E0+1 ≥ 12 inj(x).

There exists c ≥ 1 such that for any x ∈ X and v ∈ V with ‖v‖ ≤ 2−E0 ,

d(x, x exp(v)) ≤ c‖v‖c.

Recall from Lemma 7.7, that ω−1(x) inj(x) for all x ∈ X0. IncreasingE0 if necessary, we may assume that for all x ∈ X0,

c · ω(x)−cE0 < inj(x).

Since pY ≥ 1, it follows that for any y ∈ Y0 and v ∈ IZ(y,E0),

(9.3) d(y, y exp(v)) ≤ c‖v‖c ≤ c · ω(y)−cE0p−cE0Y ≤ c · ω(y)−cE0 < inj(y).

We first claim that there exists some c0 ≥ 1 so that for any y ∈ K,

(9.4) #IZ(y,E0) ≤ c0p?Y · τZ .

29

Note that as y ∈ Y0 ⊂ X0 and v ∈ IZ(y,E0), we have from (9.3) that

y exp v ∈ Z ∩X ′0.Since ω ≥ 2, for all v ∈ IZ(y,E0) we have inj(y exp v) ≥ 1

2 inj(y). Let

ρ = min1, 12 inj(y). Then

#IZ(y,E0) ·Vol(BH(y, ρ))

Vol(⋃BH(y exp v, ρ) : v ∈ IZ(y,E0) ≤ τZ

since ρ ≤ inj(y exp v) and y ∈ X0. Since inj(y)−1 ω(y) ≤ 1 + D1 ≤p?Y by Proposition 7.7 and the dimension of Z is equal to 3, we get thatVol(BH(y, ρ))−1 p?Y ; hence (9.4) follows.

Let now y ∈ Y0 − K. By taking c = 12ωs(y) in Theorem 8.1, there exists

some t := t(y) ≤ ? log(ω(y)pY ) so that

Atωs(y) < 1 +D1.

Therefore y′ := yur0at belongs to K for some r0 ∈ supp(µPSy ) ∩ [−1, 1].

Note that

sup|r|≤1,|s|≤log(ω(y)pY )†

‖v(uras)‖ ≤ c′ · (ω(y)pY )? · ‖v‖

for some absolute constant c′ ≥ 1. Since pY ≥ 1 and ω ≥ 2, for E := E0 +?,we have if ‖v‖ ≤ ω(y)−Ep−EY , then

sup|r|≤1,|s|≤log(ω(y)pY )†

‖v(uras)‖ ≤ c′ ω(y)−E+?p−E+?Y

≤ 1

2ω(y)−E0p−E0

Y ≤ ω(y′)−E0p−E0Y

using ω(y) ≥ ω(y′).Since y(exp v)ur0at = y′ exp(v(ur0at)) ∈ Z and y′ ∈ K ⊂ Y0, we thus get

thaty(exp v)ur0at = y′ exp(v(ur0at)) ∈ Z ∩X ′0.

Altogether, it follows that the map v 7→ v(ur0at) gives an injective mapfrom IZ(y,E) into IZ(y′, E0). Therefore by (9.4),

#IZ(y,E) ≤ #IZ(y′, E0) ≤ c0p?Y · τZ

as was claimed.

Definition 9.5. Define fs := fs,Y,Z : Y0 → (0,∞] by

(9.6) fs(y) :=

ω(y)sEpsEY if IZ(y) = ∅∑

v∈IZ(y) ‖v‖−s otherwise

where E > 0 as in Lemma 9.2 and IZ(y) = IZ(y,E).

Note that

(9.7) fs(y) ≥ ω(y)sEpsEY

for all y ∈ Y0 and that if Z 6= Y , we have f(y) <∞ for all y ∈ Y0.


Lemma 9.8. Let Q ⊂ H be a compact subset. Then for any y ∈ Y0 andh ∈ Q such that yh ∈ Y0, we have

fs/E(yh) ≤∑

v∈IZ(y)

‖vh‖−s/E + c0csQd

s/EQ τZp

L+sY ω(y)s

where cQ and dQ are as in §7.4, L and c0 are as in Lemma 9.2, and thesum is understood as 0 in the case when IZ(y) = ∅.

Proof. Let y ∈ Y0 and h ∈ Q with yh ∈ Y0. If IZ(yh) = ∅, then by (7.6), wehave

fs/E(yh) = ω(yh)spsY ≤ csQω(y)spsYproving the claim.

Now suppose that IZ(yh) 6= ∅. Setting

ε := ω(y)−Ep−EY /dQ,

we write

fs/E(yh) =∑

v∈IZ(yh)

‖v‖−s/E(9.9)

=∑

v∈IZ(yh),‖v‖<ε

‖v‖−s/E +∑

v∈IZ(yh),‖v‖≥ε

‖v‖−s/E .

Since by Lemma 9.2, #IZ(yh) ≤ c0pLY τZ , we have∑

v∈IZ(yh),‖v‖≥ε

‖v‖−s/E ≤(c0p

LY τZ

)ε−s/E(9.10)

≤(c0d

s/EQ τZp

L+sY

)ω(y)s.

If there is no v ∈ IZ(yh) with ‖v‖ ≤ ε, then this proves the claim by(9.9). If v ∈ IZ(yh) satisfies ‖v‖ < ε, then

‖vh−1‖ ≤ dQε = ω(y)−Ep−EY ;

in particular, vh−1 ∈ IZ(y). Therefore, by setting v′ = vh−1,∑v∈IZ(yh),‖v‖<ε

‖v‖−s/E ≤∑

v′∈IZ(y)

‖v′h‖−s/E .

Together with (9.10), this finishes the proof.

Definition 9.11 (Definition of Fs,λ). For λ ≥ 1, we define Fs,λ = Fs,λ,Y,Z :Y0 → R as follows:

(9.12) Fs,λ(y) = fs/E(y) + λ ωs(y).

Theorem 9.13. There exists 0 < λ p?Y τZ so that the following holds:there exists D p?Y τ

?Z and for every 0 < c ≤ 1/2 there exists some tc

| log c|+ log pY so that for all t ≥ tc, we have

AtFs,λ(y) ≤ c Fs,λ(y) +D for all y ∈ Y0.

31

This theorem will follow from Proposition 5.3, once we verify that theconditions in that proposition hold. Let us begin with the following log-continuity statement.

Lemma 9.14. There exists 1 ≤ σ p?Y τZ so that

σ−1Fs,λ(y) ≤ Fs,λ(yh) ≤ σFs,λ(y)

for all y ∈ Y0 and all h ∈ BH(1) so that yh ∈ Y0.

Proof. Since BH(1)−1 = BH(1), it suffices to show the inequality ≤.By Lemma 9.8, applied with c := cBH(1) and d := dBH(1), we have that

for all h ∈ BH(1) with yh ∈ Y0,

fs/E(yh) ≤∑

v∈IZ(y)

‖vh‖−s/E + c0csds/EτZp

L+sY ω(y)s

≤ ds/E∑

v∈IZ(y)

‖v‖−s/E + c0csds/EτZp

L+sY ω(y)s.

Recall that λ p?Y τZ . If IZ(y) = ∅, fs/E(y) p?Y τZωs(y); hence

Fs(yh) p?Y τZω(y)s p?Y τZ(fs/E(y) + λωs(y)) p?Y τZFs(y)

where all the implied constants are absolute constants.Suppose now that IZ(y) 6= ∅. Then

fs/E(yh) ≤ dE/sfs/E(y) + csds/EτZpL+sY ω(y)s.

Hence Fs(yh) p?Y τZFs(y); again, the implied constants are absolute andwe used the fact that λ p?Y τZ .

We also have the following lemma which is an analogue of Lemma 8.4.

Lemma 9.15. Let the notation be as in Theorem 9.13. For every 0 < c ≤1/2, there exist

• λ p?Y τZc−? and

• τ | log c|+ log pY

so that for all y ∈ Y0 and all 1 ≤ ρ ≤ 2, we have

Aτ,ρFs,λ(y) ≤ c Fs,λ(y) + λD1

where D1 > 0 is as in Theorem 8.1.

Proof. Let y ∈ Y0. Apply Theorem 8.1 with c/2. Thus, there exists τ1 log |c|+ log pY so that for all τ ≥ τ1 and 1/2 ≤ ρ ≤ 2, we have

(9.16) Aτ,ρωs(y) <

c

2ωs(y) +D1.

By Lemma 6.1, for every non-zero vector v ∈ V , we have

(9.17) sup1≤ρ≤2

1

µPSy [−ρ, ρ]

∫ ρ

−ρ

ρ

‖vurat‖s/EdµPS

y (r) ≤ c1pY e−st/(2E)‖v‖−s/E

where c1 ≥ 1 is an absolute constant.


Let τ ≥ τ1 be so thatc1pY e

−sτ/(2E) = c.

We will show that the claim holds for this τ and

λ := 2ceτs(1+1/E)τZpL+sY

where c ≥ 1 is an absolute constant which will be determined momentarily.Note that since esτ/(2E) = c1pY /c ≥ 2, we have λ ≥ 1.

As we have done before, the argument is based on comparing fs/E(yuraτ )and fs/E(y) for r ∈ (−1, 1) such that yuraτ ∈ Y0.

Let Q ⊂ H be the compact subset ur : |r| ≤ 2at : |t| ≤ τur : |r| ≤2. Then

cQeτ and dQeτ

up to an absolute multiplicative constant, where cQ and dQ are as in §7.4.Hence, by Lemma 9.8, we have that for any |r| ≤ 2,

fs/E(yuraτ ) ≤∑

v∈IZ(y)

‖vuraτ‖−s/E + c eτs(1+1/E)τZpL+sY ω(y)s

where c is an absolute constant.Use this c to define λ. We now average this over [−ρ, ρ] with respect to

µPSy . Then using (9.16) and (9.17), we get that

Aτ,ρfs/E(y) ≤ c · fs/E(y) + c eτs(1+1/E)τZpL+sY ω(y)s(9.18)

= c · fs/E(y) + λ2ω(y)s;

recall that c1pY e−sτ/(2E) = c.

Then by (9.16) and (9.18), we have

Aτ,ρFs,λ(y) = Aτ,ρfs/E(y) +Aτ,ρωs(y)

≤ c · fs/E(y) + cλ2 ω(y)s + cλ

2 ω(y)s + λD1

= c · Fs,λ(y) + λD1.

Since λ pL+s+EY τZc

−E , this proves the claim.

Proof of Theorem 9.13. In view of Lemma 9.14 and Lemma 9.15, the theo-rem follows from Proposition 5.3(1).

Corollary 9.19. Let the notation be as in Theorem 9.13. For any y ∈ Y0

we havelim sup

tAtFs,λ(y) ≤ 1 +D.

Proof. In view of Lemma 9.14 and Lemma 9.15, the theorem follows fromProposition 5.3(2).

Theorem 9.20. For any y ∈ Y0, we have

limt→∞

AtFs/2,λ(y) = mY (Fs/2,λ).

In particular, mY (Fs/2,λ) ≤ 1 +D.

33

Proof. The proof is similar to the proof of Proposition 8.6 and is based onTheorem 5.16, Theorem 9.13, and Lemma 5.14.

Recall that Z 6= Y . Hence, F•,λ : Y0 → [2,∞) is a continuous function for• = s, s/2. For every R > 0, let

FR = Fs/2,λ − Fs/2,λ · χRwhere χR is a continuous function which equals 1 on y ∈ Y0 : Fs/2,λ ≥ R+1and equals 0 on y ∈ Y0 : Fs/2,λ ≤ R. Then by Theorem 5.16, for anyy ∈ Y0, we have

(9.21) limt→∞

AtFR(y) = mY (FR).

Moreover, in view of Theorem 9.13 and Corollary 9.19, Fs/2,λ satisfies (1)and (2) in Proposition 5.3. Hence, using Lemma 5.14, we have

(9.22) |mY (Fs/2,λ)−mY (FR)| → 0 as R→∞.

We now compute that

lim supt|AtFs/2,λ(y)−mY (Fs/2,λ)| ≤ lim sup

t|At(Fs/2,λ · χR)|+

lim supt|At(FR)−mY (FR)|+ |mY (FR)−mY (Fs/2,λ)|.

The second and third term on the right side of the above tend to 0by (9.21) and (9.22). We thus estimate lim supt |At(Fs/2,λ · χR)|.

First note that

(9.23) Fs/2,λ · χR ≤ R−1(Fs/2,λ · χR)2.

Now by the Cauchy-Schwartz inequality and the definition of Fs/2,λ, we have

(Fs/2,λ · χR(y))2 p?Y τZ

ω(y)sEpsEY + λ2ωs(y) if IZ(y) = ∅∑

v∈IZ(y) ‖v‖−s + λ2ωs(y) otherwise

where we used Lemma 9.2 to control #IZ(y).Since λ p?Y τZ , we get that (Fs/2,λ·χR(y))2 p?Y τZFs,λ. This and (9.23)

now imply that

Fs/2,λ · χR p?Y τZR−1Fs,λ.

Applying the averaging operator At to the above stimate, we get that

At(Fs/2,λ · χR

) p?Y τZR

−1AtFs,λ.

Since AtFs,λ < D + 1 (see Corollary 9.19), it follows that

lim supt|At(Fs/2,λ · χR)| R−1,

where the multiplicative constant is independent of R. The first claim nowfollows by letting R→ 0.

The second claim follows from the first claim and Corollary 9.19.


10. Quantitative isolation of a closed orbit

In this section, we prove Theorem 1.4 in two installations.

Theorem 10.1 (Isolation in distance). There exist N > 0 depending on Γand an absolute constant c > 0 such that the following holds: Let Y 6= Zbelong to H. For any 0 < ε < εX and y ∈ Y0 ∩Xε, we have

d(y, Z) ≥ c · p−N/δY

Y τ−N/δYZ mY (B(y, ε))N/δ

Y .

Proof. Fix F := Fs,λ,Y,Z as in Theorem 9.13 for a fixed s = 14δY . By Lemma

9.14, there exists 0 < σ p?Y τZ so that

σ−1F (y) ≤ F (yh) ≤ σF (y)

for all y ∈ Y0 and all h ∈ BH(εX) so that yh ∈ Y0.By Theorem 9.20 and Corollary 9.19 , we have

mY (F ) ≤ 1 +D.

Let 0 < ε ≤ εX . Let y ∈ Y0 ∩ Xε; then inj(y) ≥ ε and yBH(ε) = B(y, ε).For all h ∈ BH(ε), F (y) ≤ σF (yh) and hence

F (y) ≤ σ

∫x∈yB(ε) F (x)dmY (x)

mY (B(y, ε))≤ σ mY (F )

mY (B(y, ε))≤ σ D + 1

mY (B(y, ε)).

In view of the definition of F , we get that

(10.2) fs/E,Y,Z(y) ≤ (D + 1)σ

mY (B(y, ε))− λωs(y) ≤ (D + 1)σ

mY (B(y, ε)).

It follows from the definition of fs,Y,Z that

fs/E,Y,Z(y) ≥ d(y, Z)−s/E .

Since s = δY /4,

d(y, Z) ≥(mY (B(y, ε))

(D + 1)σ

)4E/δY.

Now the claim follows since D and σ are p?Y τ?Z .

Theorem 10.3 (Isolation in measure). There exist N > 0 depending on Γand an absolute constant c > 0 so that the following holds. Let 0 < ε ≤ 1/2.For all Y,Z ∈ H with Z 6= Y , we have

mY y ∈ Y0 : d(y, Z) ≤ ε ≤ c · pNY τNZ εδY /N .

Proof. Fix s := 14δY and let F := Fs,λ,Y,Z as was done in the proof of

Theorem 10.1. For any η > 0, define

Ωη := y ∈ Y0 : F (y) > η−1;

note that if y ∈ Y0 satisfies either d(y, Z) ≤ ηE/s or ω(y) > η−1/s, theny ∈ Ωη.

35

By the definition of Ωη, we have

mY (Ωη)

η≤∫

Ωη

FdmY ≤ mY (F ).

By Theorem 9.20, mY (F ) ≤ D + 1. Hence we get that

(10.4) mY y ∈ Y0 : d(y, Z) ≤ ηE/s ≤ mY (Ωη) ≤ (D + 1)η.

Recall that εs/E = εδY /(4E) and D p?Y τ

?Z . The claim thus follows

from (10.4) applied with ε = ηE/s.

Proof of Theorem 1.4. By Corollaries 3.14 and 4.6, we have sY pY andτZ areat(SZ), respectively. Theorem 1.4 thus follows from Theorems 10.1and 10.3

11. Appendix: Proof of Theorem 1.1 in the compact case

In this section we present the proof of Theorem 1.1 when X is compact.As was mentioned in the introduction, this case is due to G. Margulis. Hisoriginal proof used the operator Aσψ(y) =

∫H ψ(yh)dσ(h) where σ is any

measure whose support generates a Zariski dense subgroup of H, but we usethe horocyclic average as in (11.4).

Let Y 6= Z be two closed orbits of H in X = Γ\G. Fix small 0 < ε0 < 1/2so that 0 < 2ε0 < minx∈X inj(x).

Define fZ : Y → [2,∞) as

fZ(y) =

∑v∈IZ(y) ‖v‖−1/3 if IZ(y) 6= ∅

ε−1/30 otherwise

where

IZ(y) = v ∈ isl2(R)− 0 : ‖v‖ ≤ ε0 & y exp(v) ∈ Z.

Remark 11.1. With the proof given in Lemma 6.1, the exponent 1/3 canbe replaced with any positive number s strictly less than 1/2, which is thereciprocal of the highest weight of the adjoint representation of H — this isneeded to ensure the validity of the convergence of the geometric series in(6.4) in Lemma 6.1. One can give a slightly different argument which allowsfor any exponent 0 < s < 1, see [7, Lemma 5.1].

We use the following special case of Lemma 6.1: For any v ∈ isl2(R) with‖v‖ = 1, we have

(11.2)

∫ 1

0

ds

‖vusat‖1/3≤ Ce−t/3

where vh = Ad(h)(v) for h ∈ H.


Remark 11.3. It is worth noting that Lemma 6.1 considers symmetricinterval [−1, 1] — this is necessary in the infinite volume setting; indeed thehalf interval [0, 1] may even be a null set for µPS

y for some y, see (3.2) forthe notation.

For t > 0 and y ∈ Y , define

(11.4) AtfZ(y) =

∫ 1

0fZ(yusat)ds.

Proposition 11.5. There exist absolute constants τ > 0 and b > 1 suchthat for all y ∈ Y ,

(11.6) AτfZ(y) ≤ 1

2fZ(y) + bVol(Z).

Proof. Let C be as in (11.2), and let τ be given by the equation Ce−τ/3 =1/2. We compare fZ(yurat) and fZ(y) for r ∈ [0, 1]. Let C1 be large enoughso that ‖vh‖ ≤ C1‖w‖ for all v ∈ isl2(R) and all

h ∈ us : s ∈ [0, 1]at : |t| ≤ τus : s ∈ [−1, 1].

Let v ∈ IZ(yusaτ ) be so that ‖v‖ < ε0/C1. Then ‖va−τu−s‖ ≤ ε0; inparticular, va−τu−s ∈ IY (y).

In the following, if IZ(·) = ∅, the sum is interpreted as to equal to ε−1/30 .

In view of the above observation and the definition of f , we have

fZ(yusaτ ) =∑

v∈IZ(yusaτ )

‖v‖−1/3

=∑

‖v‖<ε0/C1

‖v‖−1/3 +∑

‖v‖≥ε0/C1

‖v‖−1/3

≤∑

v∈IZ(y)

‖vusaτ‖−1/3 +∑

‖v‖≥ε0/C1

‖v‖−1/3(11.7)

Moreover, note that #IZ(yusaτ ) Vol(Z). Hence,

(11.8)∑‖v‖≥ε0/C1

‖v‖−1/3 Vol(Z)ε−1/30 .

We now average (11.7) over [0, 1]. Then using (11.8) and (11.2) we get

AτfZ(y) ≤ 12fZ(y) +O(Vol(Z));

as was claimed in (11.6).

Let mY be the H-invariant probability measure on Y :

Corollary 11.9.

(11.10) mY (fZ) ≤ 2bVol(Z).

Proof. SincemY is anH-invariant probability measure, mY (AτfZ) = mY (fZ).Hence the claim follows by integrating (11.6) with respect to mY .

37

Proof of Theorem 1.1. There exists σ = σ(ε0) > 0 such that for anyh ∈ BH(ε0) and y ∈ Y , fZ(y) ≤ σfZ(yh). Hence

fZ(y) ≤ σ ·

∫BH(ε0) fZ(yh)dmY (yh)

mY (B(y, ε0))

≤ σ mY (fZ)

mY (B(y, ε0))≤ σ2bε−3

0 Vol(Y ) Vol(Z).

Since dist(y, Z)−1/3 ≤ fZ(y), this shows Theorem 1.1(1):

d(y, Z) Vol(Z)−3 Vol(Y )−3.

Theorem 1.1(2) follows as

mY y ∈ Y : d(y, Z) ≤ ε ≤ mY y ∈ Y : fZ(y) ≥ ε−1/3

≤ ε1/3mY (fZ) ≤ 2bVol(Z)ε1/3.

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Mathematics Department, UC San Diego, 9500 Gilman Dr, La Jolla, CA92093

E-mail address: [email protected]

Mathematics department, Yale university, New Haven, CT 06520 and KoreaInstitute for Advanced Study, Seoul, Korea

E-mail address: [email protected]

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