ORBITAL COUNTING FOR SOME CONVERGENT GROUPS

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ANNALES DE

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Marc Peigné, Samuel Tapie & Pierre Vidotto Orbital counting for some convergent groups Tome 70, n^o3 (2020), p. 1307-1340.

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ORBITAL COUNTING FOR SOME CONVERGENT GROUPS

by Marc PEIGNÉ, Samuel TAPIE & Pierre VIDOTTO (*)

Abstract. —We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen–Margulis measure and whose Poincaré series converges at the critical exponentδΓ. We obtain an explicit asymptotic for their orbital growth function. Namely, for anyα∈]1,2[

and any smooth slowly varying function L : R → (0,+∞), we construct N- dimensional Hadamard manifolds (X, g) of negative and pinched curvature, whose group of oriented isometries possesses convergent geometrically finite subgroups Γ such that, asR→+∞,

NΓ(R) :=]{γ∈Γ|d(o, γ·o)6R} ∼CΓ(o)L(R) R^α e^δ^Γ^R, for someCΓ(o)>0 depending on the base pointo.

Résumé. —Nous construisons des variétés géométriquement finies à courbure strictement négative pincée, dont le flot géodésique possède une mesure de Bowen- Margulis non ergodique infinie, et dont la série de Poincaré converge à l’exposant δΓ, et nous obtenons une estimation précise du comportement asymptotique de la fonction orbitale de ce groupe. Plus précisément, pour toutα∈]1,2[ et toute fonction à variations lentes L:R→(0,+∞), nous construisons des variétés de Hadamard (X, g) de dimensionN>2 dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis Γ tels que, lorsqueR→+∞,

NΓ(R) :=]{γ∈Γ|d(o, γ·o)6R} ∼CΓ(o)L(R) R^α e^δ^Γ^R, oùCΓ(o) est une constante strictement positive qui dépend du pointo.

1. Introduction

We fix N > 2 and consider a N-dimensional Hadamard manifold X of negative, pinched curvature −B² 6 KX 6 −A² < 0. Without loss of

Keywords:Poincaré exponent, convergent/divergent groups, orbital function.

2020Mathematics Subject Classification:58F17, 58F20, 20H10.

(*) M. Peigné thanks theVietnam Institute for Advanced Studies in Mathematics(VI- ASM) in Ha Noi for generous hospitality and financial support during the first semester 2017, when this paper was written.

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generality, we may assumeA616B. Let Γ be aKleinian groupofX, i.e.

a discrete, torsion free group of isometries ofX, with quotientXΓ= Γ\X. In this paper, we study the asymptotic behavior of the orbital function:

NΓ(x,y;R) :=]{γ∈Γ|d(x, γ·y)6R}

for x,y ∈ X. The function NΓ has been the subject of many investiga- tions since Margulis [11]; see Roblin’s book [17] and references therein for an overview of the subject. The first step is to consider its exponential growth rate

δ_Γ = lim sup

R→∞

1

Rln(N_Γ(x,y;R));

by the triangular inequality,δΓ does not depend on the chosen base points xand y. The exponentδ_Γ coincides with the exponent of convergence of thePoincaré series associated with Γ:

P_Γ(x,y;s) :=X

γ∈Γ

e^{−sd(x,γ·y)}, x,y∈X.

The realδ_Γ is called thecritical exponent of Γ. It coincides with the topo- logical entropy of the geodesic flow (φt)_t∈R on the unit tangent bundle T¹X_Γ ofX_Γ, restricted to its non-wandering set [12].

Recall that any orbit Γ·x accumulates on a closed subset ΛΓ of the geometric boundary ∂X of X which does not depend on x. This set is called thelimit set of Γ, it contains 1, 2 or infinitely many points; in the last case, one says that Γ is non elementary. A point x ∈ ΛΓ is said to beradial when it is approached by orbit points in some M-neighborhood of any given ray issued from x, for some M > 0. The critical exponent also equals the Hausdorff dimension of theradial limit set Λ^rad_Γ of Γ with respect to some natural metric on∂X.

The group Γ is said to beconvergentifP_Γ(x,y;δ_Γ)<∞, anddivergent otherwise. Divergence can also be understood in terms of dynamic. Indeed, by Hopf–Tsuji–Sullivan theorem, it is equivalent to ergodicity and total conservativity of the geodesic flow with respect to the Bowen–Margulis measuremΓ on the non-wandering set of (φt)_t∈R in T¹XΓ (see again [17]

for a complete description of the construction ofm_Γ and for a proof of this equivalence).

The following statement is the most general one concerning the asymptotic behavior of the functionNΓ(x,y;R).

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Theorem 1.1 ([17]). — LetΓ be a non elementary, discrete subgroup of isometries ofX. ThenδΓ is a true limit. Furthermore, ifΓ has a non- arithmetic length spectrum⁽¹⁾, then it holds, asR→+∞,

(1) ifkm_Γk<∞, thenN_Γ(x,y;R)∼ ^kµ_δ^x^k.kµ^y^k

ΓkmΓk e^δ^Γ^R, (2) ifkm_Γk=∞, thenN_Γ(x,y;R) =o(e^δ^Γ^R),

where(µx)x∈X is the family of PattersonδΓ-conformal densities ofΓ, and m_Γ the Bowen–Margulis measure onT¹X_Γ.

By the Poincaré Recurrence Theorem, ifkmΓk<+∞then Γ is divergent.

When the Bowen–Margulis measure has infinite mass, Γ may be divergent or convergent.

In this paper, we study the asymptotic behavior ofNΓ(x,y;R) for a class of convergent groups Γ; thus, their Bowen–Margulis measure is infinite.

As far as we know, the only known convergent groups Γ with a precise asymptotic forN_Γ are normal subgroups ΓCΓ₀ of a co-compact group Γ₀ such that the quotient Γ0/Γ is virtually isometric to the latticeZ^kfor some k>3, see [16].

The finiteness ofmΓis not easy to establish in general. A precise criterion ensuring this finiteness for geometrically finite groups has been obtained in [6] and recently generalized to all non elementary groups in [15].

Recall that a discrete group Γ (or the quotient manifold X/Γ) is said geometrically finite if its limit set Λ_Γ decomposes into the radial limit set and the Γ-orbit of finitely manybounded parabolic pointsx1, . . . , x`, fixed respectively by some parabolic subgroupsPi,16i6`, acting co-compactly on∂X\{xi}. We refer to [3] for a complete description of geometrical finiteness in variable negative curvature. Finite volume manifoldsXΓ (possibly non compact) are particular cases of geometrically finite manifolds.

For geometrically finite groups, the orbital functions NP_i(x,y;·) of the parabolic subgroupsP_i,16i6`, contain the relevant information about the metric inside the cusps, which may implymΓ to be finite or infinite.

On the one hand, it is proved in [6] that the divergence of the parabolic subgroups P ⊂ Γ implies δP < δΓ, which yields that Γ is divergent andkmΓk <∞. On the other hand, there exist geometrically finite groups with parabolic subgroupsP satisfyingδ_P =δ_Γ: we call such groups exoticand say that the parabolic subgroupP(or the corresponding cuspsC) is dominant when δ_P = δ_Γ. Thus, dominant parabolic subgroups of exotic geometrically finite groups Γ are necessarily convergent. However, the

(1)It means that the set{`(γ) |γ ∈ Γ} of lengths of closed geodesics of XΓ is not contained in a discrete subgroup ofR.

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group Γ itself may be convergent or divergent. Explicit constructions of such groups are given in [6] and [14]; we present a similar construction in Section 2 below.

In this paper, we consider a Schottky product Γ of elementary subgroups Γ₁, . . . ,Γp+q, of isometries of X (see Section 3 for the definition) with p+q>3. Such a group is geometrically finite. We assume that Γ is convergent; thus, by [6], it is exotic and possesses factors (say Γ1, . . . ,Γp, p>1) which are dominant parabolic subgroups of Γ. We assume that, up to the dominant factor e^δ^Γ^R, the orbital functions NΓ_j(x,y;·) of these groups satisfy some asymptotic condition of polynomial decay at infinity.

Theorem 1.2. — Fixp, q∈Nsuch that p>1, p+q>2and let Γbe a Schottky product of elementary subgroupsΓ1,Γ2. . . ,Γp+q of isometries of a pinched negatively curved manifoldX, whereΓ₁, . . . ,Γ_pare parabolic.

Fix o ∈ X and assume that the metric g on X satisfies the following assumptions.

(H₁) The groupΓis convergent with Poincaré exponentδ_Γ.

(H2) There existα∈ ]1,2[, a smooth slowly varying functionL⁽²⁾ and positive constantsc₁, . . . , cpsuch that, for any16j6pand∆>0,

R→+∞lim R^α L(R)

X

γ∈Γj

R6d(o,γ·o)<R+∆

e^−δ^Γ^d(o,γ·o)=cj∆.

(H₃) For anyp+ 16j6p+q and∆>0,

R→+∞lim R^α L(R)

X

γ∈Γj

R6d(o,γ·o)<R+∆

e^−δ^Γ^d(o,γ·o)= 0.

Then, for anyx,yinX, there exists a constantC_Γ(x,y)>0such that, as R→+∞,

]{γ∈Γ|d(x, γ·y)6R} ∼CΓ(x,y) L(R) R^α e^δ^Γ^R.

Remark 1.3. — Hypothesis (H₁) (resp. (H₂)) deals with the asymptotic behavior of the convergent Poincaré seriesPΓ_j(o,o;δΓ) for 16j 6p(resp.

p+ 16j6p+q). When they are satisfied, the same properties hold for the seriesP_Γ_j(x,y;δ_Γ), for anyx,yinX, up to the following modification: for 16j 6p, the constant cj has to be replaced bycj(x,y) =e^−δB^xj^(y,x)cj

wherex_j is the fixed point of the parabolic group Γ_j.

(2)A function L is said to be “slowly varying” if it is positive, measurable and L(λt)/L(t)→1 ast→+∞for everyλ >0.

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Remark 1.4. — The constants C_Γ(x,y) can be explicited (see Equal- ity (4.21) and Proposition 4.6 for explanations when x = y = o). It is (non explicitly but) closely related to similar constant appearing in [18, Theorem C].

Remark 1.5. — In this paper we prove Theorem 1.2 whenp+q>3. In Section 3, we briefly explain how to extend the proof to the casep+q= 2.

The importance of the convergence hypothesis (H1) in the previous theorem is illustrated by the following result, previous work of one of the authors [18], which concerns the case when Γ is divergent with infinite Bowen–Margulis measure.

Theorem 1.6 ([18, Theorem C]). — Let Γ be a Schottky product of p+q>2elementary subgroups Γ1,Γ2. . . ,Γp+q of isometries of a pinched negatively curved manifoldXwithp>1. Let us fixo∈Xand assume that Γisdivergentwith infinite Bowen–Margulis measure and hypotheses(H2), (H₃)hold.

Then, for any x,yin X, there exists a positive constant C_Γ⁰(x,y)such that, asR→+∞,

]{γ∈Γ|d(x, γ·y)6R} ∼C_Γ⁰(x,y) e^δ^Γ^R R^2−αL(R).

The difference of asymptotic behaviors between Theorems 1.2 and 1.6 may seem surprising, since it is possible to vary smoothly the Riemannian metric gα,L from a divergent to a convergent case, preserving hypotheses (H₂) and (H₃), cf. [14] and Paragraph 2.3 below. Let us explain briefly the reasons of this difference; the details are given in Section 4.

Let Γ = Γ1∗ · · · ∗Γp+q be a Schottky product of elementary subgroups.

For anyγ∈Γ, γ6= Id, there existsk>1 such thatγ may be decomposed as γ = a1. . . ak, with ai ∈ Γ1∪ · · · ∪Γp and ai, ai+1,1 6 i < k, do not belong to the same subgroup Γ_j; the integerkis called the symbolic length of Γ and we denote Γ(k) the set of γ ∈ Γ with symbolic lengthk. When Γ satisfies hypotheses (H₂) and (H₃) (it is either convergent or divergent) then for allk∈N, there existsCk>0 such that asR→+∞,

]{γ∈Γ(k)|d(o, γ·o)6R} ∼Ck

L(R) R^α e^δ^Γ^R.

On the one hand, when Γ is convergent, the estimates of ]{γ ∈ Γ(k)| d(o, γ·o)}are summable and the Lebesgue dominated convergence Theo- rem, yields the asymptotic given in Theorem 1.2.

On the other hand, when Γ is divergent, these estimates are no longer summable and the asymptotic of {γ ∈ Γ|d(o, γ·o) 6 R} as R → +∞

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only depends on theγ=a₁. . . a_k withkR. A more refined estimate is needed and yields Theorem 1.6.

Remark 1.7. — In Section 2.2, we present parabolic groups satisfying Hypothesis (H₂). The condition α > 1 ensures that the parabolic groups Γ1, . . . ,Γp are convergent. The additional conditionα <2 is used in Lem- ma 4.3 to obtain a uniform upper bound for the powers of some Markov operatorPe introduced in Section 4. The proof of this lemma relies on [18], cf. Proposition 4.5, which is only valid for α ∈ (1,2). The statement of Theorem 1.2 remains open whenα>2.

Remark 1.8. — Hypothesis (H₃) is satisfied as soon as factors Γ_j,p+ 16 j6p+q, have a critical exponent δΓ_j < δΓ: thus, the quantity

X

γ∈Γj

R6d(o,γ·o)<R+∆

e^{−δd(o,γ·o)}

decreases exponentially quickly and (H₃) holds. For instance, if all the Γ_j are generated by an hyperbolic isometry ofX, their critical exponents equal all 0 and (H₃) is satisfied.

This article is organized as follows. In Section 2, we recall some back- ground on negatively curved manifolds, and we construct examples of metrics for which the hypotheses of Theorem 1.2 are satisfied.

In Section 3, we introduce theRuelle operator; this is the key analytical tool to establish the asymptotic of the orbital counting function. Eventually, Section 4 is devoted to the proof of Theorem 1.2; firstly, we assumex = y=oand then explain in Subsection 4.5 how to extend it for any points xandyinX.

2. Convergent parabolic groups and Schottky groups 2.1. Geometry of negatively curved manifolds

In the sequel, we fixN >2 and consider aN-dimensional complete con- nected Riemannian manifoldX with metric g whose sectional curvatures satisfy:−B²6KX 6−A²<0 for fixed constantsAandB.

We denote bydthe distance onX induced by the metricg.

Let ∂X be the boundary at infinity of X and z ∈ X. The family of functions (y7→d(z,x)−d(x,y))_x∈X converges uniformly on compact sets to the Busemann function Bx(z,·) when x → x ∈ ∂X. The Busemann

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function satisfies the fundamental cocycle relation: for any x ∈ ∂X and anyx, y, z inX,

Bx(x,z) =Bx(x,y) +Bx(y,z).

The Gromov product betweenx, y∈∂X×∂X,x6=y, is defined by (x|y)o= Bx(o,z) +By(o,z)

2

wherezis any point on the geodesic (x, y) joiningxto y.

We fix once for all an origin o∈X. For x∈∂X, thehoroballs H_x and thehorospheres∂Hx centered atxare respectively the sup-level sets and the level sets of the function Bx(o,·). For any t ∈ R, we set Hx(t) :=

{y/Bx(o,y) > t} and ∂Hx(t) := {y/Bx(o,y) = t}; the parameter t = Bx(o,y)− Bx(o,x) is theheightofywith respect tox. When no confusion is possible, we omit the indexx∈∂X.

By [2], the expression

D(x, y) =e^−A(x|y)^o

defines a distance on∂X satisfying the following property: for anyγ∈Γ D(γ·x, γ·y) =e⁻^A²^B^x^(γ⁻¹^·o,o)e⁻^A²^B^y^(γ⁻¹^·o,o)D(x, y).

In other words, the isometry γ acts on (∂X, D) as a conformal transfor- mation with coefficient of conformality|γ⁰(x)|o =e^−AB^x^(γ⁻¹^·o,o) at xand satisfies the following equality

(2.1) D(γ·x, γ·y) =p

|γ⁰(x)|o|γ⁰(y)|oD(x, y).

The functionx7→b(γ, x) :=Bx(γ⁻¹·o,o) plays a central role to describe the action of the isometry γ on the boundary at infinity ∂X. It satisfies the following “cocycle property”: for any isometries γ1, γ2 of X and any x∈∂X,

(2.2) b(γ₁γ₂, x) =b(γ₁, γ₂·x) +b(γ₂, x).

IfP is a parabolic group of isometries ofX(not necessarily cyclic), it fixes a unique pointx_P ∈∂X. We writep→+∞, p∈ Pfor any sequence (pn)n>1

of elements ofP such that limn→+∞d(o, pn·o) = +∞. Note that lim

p∈P

p→+∞

p·o=x_P.

The following lemma is an immediate consequence of the definition of the Busemann functions and the Gromov product. It is of great use in the sequel.

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Lemma 2.1.

(1) For any hyperbolic isometryh with repulsive and attractive fixed pointx⁻_h = lim_n→+∞h⁻ⁿ·oandx⁺_h = lim_n→+∞hⁿ·orespectively, we have

b(h^±n, x) =d(o, h^±n·o)−2(x^±_h|x)o+_x(n)

with lim_n→+∞_x(n) = 0, the convergence being uniform on the compact sets of∂X\ {x^∓_h}.

(2) For any parabolic groupP with fixed pointx_P, we have b(p, x) =d(o, p·o)−2(x_P|x)o+x(p) with lim p∈P

p→+∞x(p) = 0, the convergence being uniform on the compact sets of∂X\ {xP}.

2.2. On the existence of convergent parabolic groups In this section, we present briefly some known results about the existence of convergent parabolic groups satisfying hypothesis (H2). We refer to [14]

and [18] for the details.

We consider onR^N⁻¹×Ra Riemannian metric of the form gT =T²(t)dx²+ dt²

at point xt = (x, t) where dx² is a fixed euclidean metric on R^N−1 and T : R→R^∗+ is aC^∞ decreasing function. The group of isometries of gT

contains all the translations (x, t)7→(x+~τ , t) onR^N⁻¹×Rfixing the last coordinate.

By [4, Chapter 8, Section 3], the sectional curvature atx= (x, t) equals

• Kg_T(t) =− _T^T_(t)⁰⁰^(t) on any plane _∂

∂X_i,_∂t^∂

,16i6N−1;

• −K_g²

T(t) on any plane ∂

∂X_i,_∂X^∂

j

,16i < j 6N−1

Note that g_T has negative curvature if and only if T is convex. When T(t) = e^−t, this provides a model of the hyperbolic space of constant curvature−1.

Let us consider the decreasing function

(2.3) u:

(R^∗+→R s7→T⁻¹ ¹_s

which satisfies the following implicit equationT(u(s)) = ¹_s.This functionu is of interest since it gives precise estimates of the distance between points

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lying on the same horosphereHt :={(x, t) : x∈ R^N⁻¹}, t∈ R. Namely, the distance between xt = (x, t) and yt = (y, t) for the metric T²(t)dx² induced byg_T onH_tis equal to T(t)kx−yk. Hence, this distance equals 1 whent=u(kx−yk) and for this value oft, the union of the 3 segments [x₀,xt],[xt,yt] and [yt,y₀] lies at a bounded distance of the hyperbolic geodesic joining x0 and y0 (see [6, Lemma 4]) : this readily implies that d(x0,y0)−2u(kx−yk) is bounded.

Now, let P be a discrete group of isometries of R^N−1spanned by klin- early independent translationsp~τ₁, . . . , p~τ_kinR^N−1, with respective vectors

~

τ₁, . . . , ~τ_k.

For any n = (n1, . . . , nk) ∈ Z^k, we set ~n = n1~τ1 +· · ·+nk~τk. The translationsp_~_nare isometries of (R^N, gT) and the corresponding Poincaré series ofP is given by, up to finitely many terms,

(2.4) P_P(s) = X

k~nk>sα

e^−sd(o,p^~ⁿ^·o)= X

k~nk>sα

e^−2su(k~^nk)+sO(1).

Hence, it is sufficient to choose the function uin a suitable way in order that the seriesPP(s) converges at its critical exponent.

We present here two main explicit examples. Example 2.2 comes from [6]

where the existence of convergent parabolic groups appeared for the first time. Example 2.3, where we assumeN = 2 to simplify, is a refinement of Example 2.2.

Example 2.2. — For anyα>0, let us consider the increasingC²-function u=u_αfromR^∗+ toRsuch that

uα(s) =

(lns if 0< s61 lns+αln lns ifs>s_α.

We denoteT_α the function associated with u_α by relation (2.3); the con- stantsα>1 is chosen in such a way the metric

gα=T_α²(t)dx²+ dt²

onR^N−1×R has pinched negative curvature onX, bounded from above by−A².

In this case, Estimation (2.4) of the Poincaré series ofP becomes, up to finitely many terms,

P_P(s) = X

k~nk>s_α

e^sO(1) k~nk^2s lnk~nk^2sα.

Hence, the Poincaré exponent ofP equals k/2 andP is convergent if and only ifα > _k¹.

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Example 2.3. — Without loss of generality, we assume N = 2; indeed, the metrics gT are totally geodesics, hence the geodesic segments [o, p·o], p ∈ P, are included in the vertical plane passing through o and p·o and the computation of the distance d(o, pⁿ ·o) is a 2-dimensional problem.

We fix α > 1 and a smooth slowly varying function L : [0,+∞[ → (0,+∞). For any realt greater than somea>0 to be chosen⁽³⁾, let us set

T(t) =Tα,L(t) =e^−t t^α L(t).

We denote by uα,L the function related to Tα,L by the implicit equation (2.3) andgα,Lthe associated metric onR². We denotepthe isometry (x, t)7→(x+ 1, t) onR².

The following statement corresponds to [18, Lemma 2.2.3 and Proposi- tion 2.2.4], where explicit examples of functionsLare also given.

Proposition 2.4. — Forslarge enough,

u(s) = lns+αln lns−lnL(lns) +(s) with(s)→0ass→+∞.

Consequently, the parabolic group P = hpi on (R², gα,L) satisfies the following property: for anyn∈Nlarge enough,

d(o, pⁿ·o) = 2 lnn+ αln lnn−lnL(lnn) +(n) withlim_n→+∞(n) = 0.

In particular, for α >1, the groupP is convergent with respect tog_α,L and satisfies the assumption(H2).

The improvement between Example 2.2 and Example 2.3 relies on the observation that the cylinder R²/P endowed with the metric gα,L is a surface of revolution. This allows to use the Clairaut’s relations (see for instance [5, Section 4.4, Example 5]) to estimate the distance betweeno andpⁿ·o; computations are detailed in [18, Section 2.2.3].

2.3. On the existence of non elementary exotic groups Explicit constructions of exotic groups, i.e. non-elementary groups Γ con- taining a parabolic P whose Poincaré exponent equals δ_Γ, have been detailed in several papers; first in [6], then in [8], [14], and [18]. Let us describe them in the context of the metricsg=g_α,Lpresented above.

(3)For technical reasons, we assume in particulara>4α, see [18].

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For any a >0 andt∈R, we write Tα,L,a=

(e^−t ift6a e^−aTα,L(t−a) ift>a,

whereT_α,Lis defined in the previous paragraph. As in [14], we consider the metric onR²given bygα,L,a=T_α,L,a² (t)dx²+ dt². It is a complete smooth metric, with pinched negative curvatures, and with constant curvature−1 on R×(−∞, a). Note that gα,L,0 = gα,L and g_α,L,+∞ is the hyperbolic metric on H². From the previous subsection, it follows that for any a ∈ (0,+∞) and any τ ∈ R^∗, a parabolic group of the form P = h(x, t) 7→

(x+τ, t)i is convergent for the metric gα,L,a. This allows to reproduce the construction of a non-elementary group given in [6] and [14]; let us we present it.

Let hbe a hyperbolic isometry of H² and pbe a parabolic isometry in Schottky position with h(see next section for a precise definition). They generate a free group Γ =hh, piwhich acts discretely without fixed point onH². Up to a global conjugacy, we can suppose thatpis given by (x, t)7→

(x+τ, t) for someτ ∈R^∗. The surfaceS=H²/Γ has a cusp, isometric to R/τZ×(a₀,+∞) for some a₀ >0. Therefore, we can replace in the cusp the hyperbolic metric by gα,L,a for any a >a0; we also denote bygα,L,a

the lift ofg_α,L,ato R².

For any n ∈ Z^∗, the group Γn = hhⁿ, pi acts freely by isometries on (R², gα,L,a). It is shown in [6] that, forn >0 large enough, the group Γn

also converges. This provides a family of examples for Theorem 1.2, since by Remark 1.8 assumption (H3) is automatically satisfied.

By [14], if Γ_n is convergent for some a₀ > 0, then there exists a^∗ >

a0 such that for any a ∈ [a0, a^∗), the group Γn acting on (R², gα,L,a) is convergent, whereas fora > a^∗, it has finite Bowen–Margulis measure and hence diverges.

In the case when a=a^∗, the group Γ also diverges butm_Γ has infinite measure forα∈]1,2[ (see [14]). A precise estimate of the functionNΓdoes also exist in this case (cf. Theorem 1.6).

Remark 2.5. — In [8], the authors propose another approach based on a “strong” perturbation of the metric inside the cusp. Starting from aN- dimensional finite volume hyperbolic manifold with cuspidal ends, they modify the metric far inside one end in such a way that the corresponding parabolic group is convergent with Poincaré exponent >1 and turns the fundamental group of the manifold into a convergent group. In this construction, the sectional curvature of the new metric along certain planes

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is<−4 far inside the modified cusp. With this construction, we may extend the validity of t Theorem 1.2 tofinite volume manifolds with infinite Bowen–Margulis measure. We will not do it here.

2.4. Schottky groups

Let us fix two integers p > 1 and q > 0 such that ` := p+q > 2 and consider ` elementary groups Γ1, . . . ,Γ` of isometries of X. For all j∈ {1, . . . , `}, we write Γ^∗_j = Γ_j\{Id}; similarly Γ^∗= Γ\{Id}.

The elementary groups Γ1, . . . ,Γ` are said to be inSchottky position if there exist disjoint closed setsF_j in∂X such that, for any 16j6` (2.5) Γ^∗_j(∂X\Fj)⊂Fj.

We setF=S

jFj.

The group Γ = hΓ₁, . . . ,Γ_`i spanned by the Γ_j,1 6j 6`, is called the Schottky productof the Γj’s.

In this section, we present general properties of such Schottky groups.

We emphasize on the coding of the elements of the group by words over a countable alphabet, which is crucial for the proof of Theorem 1.2. We do not require yet that conditions (H₁), (H₂) and (H₃) hold: these hypotheses are only needed in the last section of this paper.

Property (2.5) and Klein’s tennis table criterion imply that Γ is the free product of the groups Γi:

Γ = Γ₁?Γ₂?· · ·?Γ_`.

Hence, anyγ∈Γ^∗can hence be uniquely written as the product γ=a1. . . ak

for someaj∈S

Γ^∗_j with the property that no two consecutive elementsaj

belong to the same group Γj. The setA=SΓ^∗_j is called thealphabetof Γ, anda1, . . . , ak the lettersof γ. The number kof letters ofγ =a1. . . ak is thesymbolic length ofγ; we denote by Γ(k) the set of elements of Γ with symbolic lengthk. The last letter ofγplays a special role; the index of the group it belongs to is denoted by lγ. Applying Property 4.1.3 from [18], one gets

Proposition 2.6. — There exists a constantC >0 such that for any γ∈Γ =?iΓi and anyx∈∂X\Fl_γ,

d(o, γ·o)−C6Bx(γ⁻¹·o,o)6d(o, γ·o).

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This proposition implies in particular the following crucial contraction property [1].

Proposition 2.7. — There exist a real number r ∈ ]0,1[ and C > 0 such that, for anyγ with symbolic length n >1 and any x belonging to the set∂X\Fl_γ, it holds

|γ⁰(x)|6Crⁿ.

The coding of elements of Γ extends to a coding of a large subset of ΛΓ; let us restate Proposition 1 of [13].

Proposition 2.8. — Denote by Σ⁺ the set of sequences (an)n>1 for which each letteran belongs to the alphabet A=SΓ^∗_i and such that no two consecutive letters belong to the same group (these sequences are called admissible). Fix a pointx0in ∂X\F. Then

(1) For anya= (a_n)_n_>₁∈Σ⁺, the sequence(a₁. . . a_n·x0)_n_>₁converges to a pointπ(a)in the limit set ofΓ, independent on the choice ofx0. (2) The mapπ: Σ⁺→Λ_Γ is one-to-one andπ(Σ⁺)is contained in the

radial limit set ofΓ.

(3) The complement ofπ(Σ⁺)in the limit set of Γ equals theΓ-orbit of the union of the limits setsΛΓi.

Hence, up to a countable set of points, the limit set ΛΓ of Γ coincides withπ(Σ⁺). Note that the set Λ_Γ is contained inF =S

jF_j, by (2.5).

For any 1 6 i 6`, let Λi = ΛΓ∩Fi be the closure of the set of limit points with first letter in Γ_i (not to be confused with the limit set of Γ_i, which is a finite set). Since theFi are closed disjoint subsets, we get the following useful description of ΛΓ:

(a) ΛΓ is the finite union of the sets Λi,

(b) the closed sets Λi,16i6`,are pairwise disjoint,

(c) each of these sets is partitioned into a countable number of closed disjoint subsets:

Λi= [

a∈Γ^∗_i

∪_j6=i a.Λj .

Let us fix a pointx0∈∂X\F. There exists a one-to-one correspondence between Γ·x0 and Γ. Furthermore, the point γ·x0 ∈ Fj for any γ ∈ Γ^∗ with first letter in Γ_j. We setΣe₊= Σ⁺∪Γ and notice that, by the previous Proposition, the natural mapπ:Σe+ →ΛΓ∪Γ·x0is one-to-one with image π(Σ⁺)∪Γ·x₀. Thus we introduce the following notations:

(a) Λ = Λe Γ∪Γ·x0;

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(b) Λei=ΛeΓ∩Fifor any 16i6`.

The set Λ is the disjoint union ofe {x0} and the sets Λei,1 6 i 6 `.

Furthermore, each Λe_i is partitioned into a countable number of subsets with disjoint closures:

Λei= [

a∈Γ^∗_i

a



 [

j6=i

Λej∪ {x0}



.

The cocycle b defined in (2.2) plays a central role in the sequel. Now, we introduce an extension of this cocycle (denoted alsob) which allows to control the distance (and not the value of the Buseman function) between two points of the orbit Γ·o. It is defined on Λ as follows: for anye γ ∈Γ andx∈Λ,e

b(γ, x) :=

(B_x(γ⁻¹o,o) ifx∈Λ_Γ;

d(γ⁻¹·o, g·o)−d(o, g·o) ifx=g·x0 for someg∈Γ.

The cocycle equality (2.2) is still valid for this extended functionb; furthermore,

b(γ, x₀) =d(o, γ·o).

3. Ruelle operators for Schottky groups

LetX be a Hadamard manifold with origino∈X, and Γ = Γ1?· · ·?Γ`

a Schottky product of elementary groups of isometries ofX, as defined in the previous section.

3.1. Orbital counting function and Ruelle operators Let us decompose the orbital counting function according to the symbolic length of the elements of Γ, then introduce in a natural way the main analytical tool of the proof: the Ruelle operators.

We writeδ=δ_Γ andN_Γ(o;R) =N_Γ(o,o;R). For allR >0, N_Γ(o;R) =X

γ∈Γ

X

n∈N

1[n,n+1[(R−d(o, γ·o))

=e^δRX

γ∈Γ

e^{−δd(o,γ·o)}X

n∈N

en(R−d(o, γ·o))

=e^δRX

n∈N

X

γ∈Γ

e^−δb(γ,x⁰⁾e_n(R−b(γ, x₀))

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where for alln∈Nand allt∈R,

(3.1) en(t) =e^−δt1^[n,n+1[. Thus, the quantityNΓ(o;R) decomposes as (3.2) NΓ(o;R) =e^δRX

n∈N

X

k∈N

X

γ∈Γ(k)

e^−δb(γ,x⁰⁾en(R−b(γ, x0)) where Γ(k) is the set of elements of Γ with symbolic lengthk.

For all φ∈L^∞(eΛ), allu:R→R with compact support, all s >0 and all (x, t)∈∂X×R, let us set

Le_s(φ⊗u)(x, t) = X

γ∈Γ(1)

1x /∈Λ˜_lγe^−sb(γ,x)φ(γ·x)u(t−b(γ, x)).

This is a finite sum as soon asuhas compact support.

By the cocycle property ofb, the iteratesLe^k_s of the operatorsLesmay be written as follows

Le^k_s(φ⊗u)(x, t) = X

γ∈Γ(k)

1x /∈Λ˜_lγe^−sb(γ,x)φ(γ·x)u(t−b(γ, x)) (more detailed explanations are given in the next section for the iterates of the classical Ruelle operators).

Thus, equation (3.2) may be rewritten as (3.3) NΓ(o;R) =e^δRX

n∈N

X

k∈N

Le^k_δ(1Λ^˜⊗en) (x0, R).

These operators Le_s can be seen as an R-extension of the well known class ofRuelle operators Lsassociated with the Schottky group Γ, defined formally by: for any functionφ∈L^∞(eΛ) and anyx∈Λ,e

(3.4) Lsφ(x) = X

γ∈Γ(1)

1x /∈Λ˜_lγe^−sb(γ,x)φ(γ·x)

These operators Ls are classical tools in hyperbolic dynamic, in particular to study the geodesic flow on T¹X_Γ, cf. for instance [1], [7] or [18].

In the present paper, we do not develop further considerations about the geodesic flow since we will not use it to estimate the orbital function.

3.2. Poincaré series versus Ruelle operators

From now on, we assume that all the Poincaré series of the Γj, j = 1, . . . , `, do converge at δ = δ_Γ. For instance, this condition holds when hypotheses (H1), (H2) and (H3) are satisfied.

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Definition (3.4) may be rewritten as Lsφ(x) =

`

X

j=1

X

γ∈Γ^∗_j

1x /∈Λ˜_je^−sb(γ,x)φ(γ·x)

For any 16j6`, the sequence (γ·o)_γ∈Γ_j accumulates on the fixed point(s) of Γj; hence, the sequence (b(γ, x)−d(o, γ.o))_γ∈Γ

j is bounded, uniformly inx /∈Λej. Therefore, since the Poincaré series of the Γj does converge at δ= δΓ, the quantity Lsφ(x) is well defined as soon as φ is bounded and s>δ:= max{δ_Γ_j|16j6`}.

For anys>0 andγin Γ^∗, letws(γ,·) be theweight functiondefined on Λ by: for anye s>δandγ∈Γ,

w_s(γ, x) :=











1 ifγ= Id,

e^−sb(γ,x) ifx∈Λe_j, j6=l_γ,

0 otherwise.

Observe that these functions are continuous onΛ ande Lsφ(x) = X

γ∈Γ(1)

ws(γ, x)φ(γ·x).

By a straightforward computation, one may thus check thatLsis a bounded linear operator on (C(eΛ),| · |_∞) whens>δ; we denote byρ_s(∞) its spectral radius on this space.

Let us now compute the iterates L^k_s, k > 1, of the operators L_s. The functions ws(γ,·) also satisfy the following cocycle relation: if γ1, γ2 ∈ A do not belong to the same group Γj, then

ws(γ1γ2, x) =ws(γ1, γ2·x)ws(γ2, x).

Due to this cocycle property, we may write, for anyk >1, any bounded functionφ:Λe →Rand anyx∈Λ,e

L^k_sφ(x) = X

γ∈Γ(k)

w_s(γ, x)φ(γ·x)

= X

γ∈Γ(k)

1x /∈Λ˜_lγe^−sb(γ,x)φ(γ·x).

By the “ping-pong dynamic” between the subgroups Γj,1 6 j 6 `, and Proposition 2.6, the differenceb(γ, x)−d(o, γ·o) is bounded uniformly in k>0, γ ∈Γ(k) and x /∈Λel_γ. Consequently, there exists a constantC >0

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such that, for anyx∈Λ, anye k>1 and anys>δ, L^k_s1(x)^c X

γ∈Γ(k)

e^{−sd(o,γ·o)}

wherecis a positive constant andA^c B means ^A_c 6B6cA. Hence, (3.5) PΓ(s) :=X

γ∈Γ

e^{−sd(o,γ·o)}= +∞ ⇐⇒ X

k>0

L^k_s1(x) = +∞.

In particular

(3.6) δΓ = sup{s>δ|ρs(∞)>1}= inf{s>δ|ρs(∞)61}.

We prove in the next paragraph that Γ is convergent if and only if ρ_δ(∞)<1.

3.3. On the spectrum of the operators L_s, s>δ

Following the classical approach to study the spectrum of the transfer op- eratorsLs(see [1], [7] and comments therein), we consider their restriction to the space Lip(eΛ) of Lipschitz functions fromΛ toe Cdefined by

Lip(eΛ ={φ∈C(eΛ)| kφk=|φ|_∞+ [φ]<+∞}

where[φ] = sup

06i6l

sup

x,y∈Λ˜j

x6=y

|φ(x)−φ(y)|

D(x, y)

is the Lipschitz coefficient ofφon (∂X, D).

The space (Lip(Λ),e k · k) is a Banach space and by Arzela–Ascoli Theo- rem, the identity map from (Lip(eΛ),k · k) into (C(eΛ),| · |_∞) is compact. It is proved in [1] that the operatorsLs, s>δ, act both on (C(ΛΓ),| · |∞) and (Lip(ΛΓ),k · k). P. Vidotto has extended in [18] this property to the Banach spaces (C(eΛ),| · |_∞) and (Lip(eΛ),k · k). We denote byρ_sthe spectral radius ofLson Lip(Λ). The following proposition gathers the spectral propertiese of theLswhich we need in the present paper.

Proposition 3.1. — Assume that`=p+q>3. For anys>δ, (1) ρ_s=ρ_s(∞);

(2) ρsis a simple eigenvalue ofLsacting onLip(eΛ)and the associated eigenfunctionhsis positive onΛ;e

(3) there exists06r <1 such that the rest of the spectrum ofL_s on Lip(eΛ)is included in a disc of radius6rρs.

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Sketch of the proof. — We refer to [1] and [18] for more details. In [1], it is proved that the restriction of the functionsws(γ,·), γ ∈Γ, to the set Λ_Γ belong to Lip(Λ_Γ) and that for anys >δ there exists C =C(s)>0 such that, for anyγin Γ^∗

kws(γ,·)k6Ce^{−sd(o,γ.o)}.

In [18, Proposition 8.3.1], P. Vidotto has proved that the same inequality holds for the functionsw_s(γ,·) onΛ. Thus, the operatore Lsis bounded on Lip(eΛ) whens>δ.

In order to describe its spectrum on Lip(Λ), we first write a “contractione property” for the iterated operatorsL^k_s: for allx, y∈Λ, alle k∈N and all s>δ, it holds

|L^k_sφ(x)− L^k_sφ(y)|

6 X

γ∈Γ(k)

|ws(γ, x)| |φ(γ·x)−φ(γ·y)|+ X

γ∈Γ(k)

[ws(γ,·)]|φ|_∞D(x, y).

By Proposition 2.7 and the mean value relation (2.1), there existC >0 and 06r <1 such that D(γ·x, γ·y)6Cr^kD(x, y) wheneverx, y∈Λej, j6=l_γ andγ∈Γ(k). Hence, we get

(3.7) [L^k_sφ]6rk[φ] +Rk|φ|_∞ whererk = Cr^k

|L^k_s1|∞ andRk =P

γ∈Γ(k)[ws(γ,·)]. Observe that lim sup

k

r^1/k_k =rlim sup

k

|L^k_s1|^1/k_∞ =rρ_s(∞)

whereρ_s(∞) is the spectral radius of the positive operatorL_sonC(eΛ). By Hennion’s work [10], Inequality (3.7) implies thatLsis quasi-compact and its essential spectral radius on Lip(eΛ) is less thanrρs(∞). In other words, any spectral value of L_s with modulus strictly larger than rρ_s(∞) is an eigenvalue with finite multiplicity and is isolated in the spectrum ofLs.

This implies in particularρ_s=ρ_s(∞). Indeed, the inequalityρ_s>ρ_s(∞) is obvious since the function 1 belongs to Lip(eΛ). Conversely, the strict inequality would imply the existence of a function φ ∈ Lip(eΛ) such that L_sφ=λφfor someλ∈Cof modulus > ρ_s(∞); this yields |λ||φ|6L_s|φ|

so that|λ|6ρs(∞): a contradiction.

It remains to control the nature of the value ρs in the spectrum of Ls. By the previous argument, we know that ρ_s is an eigenvalue of Ls with (at least) one associated eigenfunctionhs>0. This function is positive on Λ: otherwise, there exist 1e 6 j 6 p+q and a point y₀ ∈ Λe_j such that hs(y0) = 0. The equalityLshs(y0) =ρshs(y0) implieshs(γ·y0) = 0 for any

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γ ∈Γ with last letter 6=j. The minimality of the action of Γ on Λ_Γ and the fact that Γ·x0accumulates on Λ implieshs= 0 onΛ: a contradiction.e In order to prove thatρ_sis a simple eigenvalue on Lip(eΛ), let us introduce the so-called “Doob transform” ofLs. For anys>δ, we denote byPsthe operator defined formally by: for any bounded Borel function φ: Λe →C andx∈Λ,e

Psφ(x) = 1

ρshs(x)L(hsφ)(x) = 1 ρhs(x)

X

γ∈Γ(1)

1x /∈Λ˜_lγe^−sb(γ,x)h(γ·x)φ(γ·x).

The iterates ofP_s are given by:P_s⁰= Id and fork>1 (3.8) P_s^kφ(x) = 1

ρ^k_shs(x) X

γ∈Γ(k)

1x /∈Λ˜_lγe^−sb(γ,x)h(γ·x)φ(γ·x).

The operator Ps acts on Lip(eΛ) as a Markov operator, i.e. Psφ >0 if φ > 0 and P_s1Λ^˜ = 1Λ^˜. It inherits the spectral properties of L_s and is in particular quasi-compact with essential spectral radius <1. The spectral value 1 is an eigenvalue and it remains to prove that the associated eigenspace isC·1. Let f ∈Lip(eΛ) such that P_sf =f and 1 6j 6p+q andy0 ∈Λej such that |f(y0)|=|f|_∞. An argument of convexity applied to the inequalityP_s|f| 6 |f| readily implies |f(y₀)| =|f(γ·y₀)| for any γ∈Γ with last letter6=j. By minimality of the action of Γ onΛ, it followse that the modulus of f is constant on Λ. Applying again an argument ofe convexity, the minimality of the action of Γ onΛ and the fact that Γe ·x₀ accumulates on Λ, one proves thatf is in fact constant on Λ. Therefore,e the eigenspace ofL_sassociated withρ_sequals C·h_s.

A similar proof, using the fact that ` >3, shows that ρs is the unique eigenvalue with modulusρs; the argument is detailed in [1, Proposition III.4]

and [18, Proposition 8.3.2].

Whenp+q= 2, an analogous property holds (see [7, Lemma VII.6] for the details):

Proposition 3.2. — Assume that`=p+q= 2. For anys>δ, (1) ρs=ρs(∞);

(2) ρsis a simple eigenvalue ofLsacting onLip(eΛ)with an associated eigenfunctionh⁺_s positive onΛ =e Λe₁∪Λe₂;

(3) −ρsis a simple eigenvalue ofLsacting onLip(eΛ), with an associated eigenfunctionh⁻_s which is positive onΛe1 and negative onΛe2; (4) there exists06r <1 such that the rest of the spectrum ofL_s on

Lip(eΛ)is included in a disc of radius6rρs.