Equidistribution and Counting for Orbits of Geometrically Finite Hyperbolic Groups

Home , Hee Oh

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 2, April 2013, Pages 511–562 S 0894-0347(2012)00749-8 Article electronically published on October 2, 2012

EQUIDISTRIBUTION AND COUNTING FOR ORBITS OF GEOMETRICALLY FINITE HYPERBOLIC GROUPS

HEE OH AND NIMISH A. SHAH

Contents 1. Introduction 511 2. Transverse measures 518 Gr Leb 3. Equidistribution of ∗μE 527 4. Geometric ﬁniteness of closed totally geodesic immersions 535 5. On the cuspidal neighborhoods of Λbp(Γ) ∩ ∂S˜ 538 PS 6. Parabolic corank and criterion for ﬁniteness of μE 544 7. Orbital counting for discrete hyperbolic groups 546 8. Appendix: Equality of two Haar measures 558 Acknowledgements 560 References 561

1. Introduction 1.1. Motivation and overview. Let G denote the identity component of the special orthogonal group SO(n, 1), n ≥ 2, and V a ﬁnite-dimensional real vector space on which G acts linearly from the right. A discrete subgroup of a locally compact group with ﬁnite covolume is called a lattice. For v ∈ V and a subgroup H of G,letHv = {h ∈ H : vh = v} denote the stabilizer of v in H. A subgroup H of G is called symmetric if there exists a non-trivial involutive automorphism σ of G such that the identity component of H is the same as the identity component of Gσ = {g ∈ G : σ(g)=g}. ∈ Theorem 1.1 (Duke-Rudnick-Sarnak [9]). Fix w0 V such that Gw0 is symmetric. · Let Γ be a lattice in G such that Γw0 is a lattice in Gw0 . Then for any norm on V , #{w ∈ w Γ:w

Received by the editors April 7, 2011 and, in revised form, January 27, 2012, and May 31, 2012. 2010 Mathematics Subject Classification. Primary 11N45, 37F35, 22E40; Secondary 37A17, 20F67. Key words and phrases. Geometrically finite hyperbolic groups, mixing of geodesic flow, totally geodesic submanifolds, Patterson-Sullivan measure. The first author was supported in part by NSF Grants #0629322 and #1068094. The second author was supported in part by NSF Grant #1001654.

c 2012 American Mathematical Society Reverts to public domain 28 years from publication 511

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{ ∈ } \ where BT := w w0G : w

1.2. Statement of main result. Our generalization of Theorem 1.1 for discrete subgroups which are not necessarily lattices involves terms which can be best ex- plained in the language of hyperbolic geometry. Let Γ orbit w0Γ in each case.

When Gw0 is a symmetric subgroup associated to an involution σ,choosea Cartan involution θ of G which commutes with σ, and let o ∈ Hn be such that its ˜ stabilizer Go is the ﬁxed group of θ.ThenS := Gw0 .o is an isometric imbedding

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of Hk in Hn for some 0 ≤ k ≤ n − 1, where the embeddings of H0 and H1 mean a point and a complete geodesic, respectively. Let E˜ ⊂ T1(Hn) be the unit normal bundle of S˜. ∈ Hn InthecasewhenGRw0 is parabolic, we ﬁx any o .IfN is the unipotent ˜ ˜ ⊂ 1 Hn radical of GRw0 ,thenS := N.o is a horosphere. We set E T ( )tobethe unstable horosphere over S˜. Now in either case, we deﬁne the following Borel measure on E˜:

PS δβv+ (x,π(v)) + dμE˜ (v):=e dνx(v )

n for x ∈ H ,andβξ(x1,x2) denotes the value of the Busemann function, that is, the signed distance between the horospheres based at ξ, one passing through x1 and the other through x (see (2.2)). This deﬁnition of μPS is independent of the 2 E˜ n choice of x ∈ H . Due to the Γ-invariance property of the conformal density {νx}, ˜ PS it induces a measure on E := p(E) which we denote by μE . Fix any X0 ∈ E˜ based at o, and let A = {ar : r ∈ R} be a one-parameter subgroup of G consisting of R-diagonalizable elements such that r → a .X is a unit ∼ r 0 speed geodesic. Note that A is contained in a copy of SO(2, 1) = PSL(2, R)such r/2 −r/2 that ar corresponds to dr =diag(e ,e ). Any irreducible representation of PSL(2, R) is given by the standard action of SL(2, R) on homogeneous polynomials of degree k in two variables such that the action of −I is trivial, so k is even and (k/2)r the largest eigenvalue of dr is e . Therefore, if λ denotes the log of the largest eigenvalue of a1 on R-span(w0G), then λ ∈ N.Weset

λ −λr w0 := lim e w0ar =0, by [13, Lemma 4.2]. r→∞

Theorem 1.2. Let Γ

#{w ∈ w Γ:w 0 (Proposition 6.7), and hence the limit (1.1) is strictly positive. (3) The description of the limit changes if we do not assume the Go-invariance of the norm ·; see Theorem 7.8, Remark 7.9(3)-(5), and Theorem 7.10. | PS| ∞ (4) If Gw0 is symmetric and Γ is Zariski dense in G, then the condition μE < implies that w0Γ is discrete, for by Theorem 2.21 and Remark 2.22, w0Γ is closed in w0G, and by [13, Lemma 4.2], w0G is closed in V . Therefore w0Γisclosedand hence discrete in V . | PS| ∞ (5) If GRw0 is parabolic, then the condition μE < implies that w0Γis discrete. To see this, note that if the horosphere S˜ is based at ξ,then∂S˜ = {ξ}, n and by Theorem 2.21, ΓS˜ is closed in H and w0Γ is closed in w0G = w0G {0}. If w0Γ is not closed in V , w0γi → 0forasequence{γi}⊂Γ. Then γio → ξ

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Figure 1. An externally Γ-parabolic vector

and ξ is a horospherical limit point of Γ. Since |mBMS| is finite, the geodesic flow is mixing (Theorem 3.2), and hence by [7, Thm. A and Prop. B], ΓS˜ is dense in − π({u : u ∈ Λ(Γ)}), a contradiction to ΓS˜ being closed. Therefore w0Γ is closed and hence discrete in V . Thanks are due to the referee for the last two remarks. A discrete group Γ is called geometrically finite if the unit neighborhood of its convex core1 has finite Riemannian volume (see also Theorem 4.6). Any discrete group admitting a finite sided polyhedron as a fundamental domain in Hn is geometrically finite. Sullivan [36] showed that |mBMS| < ∞ for all geometrically finite Γ. However, Theorem 1.2 is not limited to those, as Peigné [27] constructed a large class of geometrically infinite groups admitting a finite Bowen-Margulis-Sullivan measure. We will provide a general criterion on the finiteness of skΓ(w0) in Theorem 1.14. For the sake of concreteness, we first describe the results for the standard representation of G.

1.3. Standard representation of G. Let Q be a real quadratic form of signature (n, 1) for n ≥ 2andG the identity component of the special orthogonal group SO(Q). Then G acts on Rn+1 by the matrix multiplication from the right, i.e., n+1 the standard representation. For any non-zero w0 ∈ R , up to conjugation and − commensurability, Gw0 is SO(n 1, 1) (resp. SO(n)) if Q(w0) > 0(resp.ifQ(w0) < 0). If Q(w0) = 0, the stabilizer of the line Rw0 is a parabolic subgroup. Therefore n+1 Theorem 1.2 is applicable for any non-zero w0 ∈ R ,providedskΓ(w0) < ∞ (in this case, λ =1). An element γ ∈ Γ is called parabolic if there exists a unique fixed point of γ in n n ∂H .Forξ ∈ ∂H ,wedenotebyΓξ the stabilizer of ξ in Γ and call ξ a parabolic fixed point of Γ if ξ is fixed by a parabolic element of Γ.

Noting that Gw0 is the isometry group of the codimension one totally geodesic ˜ subspace, say, Sw0 ,whenQ(w0) > 0, we give the following: n+1 Deﬁnition 1.4. Let w0Γ be discrete. Then w0 ∈ R is said to be externally ∈ ˜ Γ-parabolic if Q(w0) > 0 and there exists a parabolic ﬁxed point ξ ∂Sw0 for Γ ∩ ˜ ⊂ Hn ˜ such that Gw0 Γξ is trivial, where ∂Sw0 ∂ denotes the boundary of Sw0 in Hn.

1 n n The convex core CΓ ⊂ Γ\H of Γ is the image of the minimal convex subset of H which contains all geodesics connecting any two points in the limit set of Γ.

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3 For n =2,w0 ∈ R with Q(w0) > 0 is externally Γ-parabolic if and only if the ˜ \Hn projection of the geodesic Sw0 in Γ is divergent in both directions, and at least ˜ H2 one end of Sw0 goes into a cusp of a fundamental domain of Γ in (see Figure 1).

Theorem 1.5 (On the ﬁniteness of skΓ(w0)). Let Γ be geometrically ﬁnite and w0Γ be discrete.

(1) If δ>1,thenskΓ(w0) < ∞. (2) If δ ≤ 1,thenskΓ(w0)=∞ if and only if w0 is externally Γ-parabolic.

Corollary 1.6. Let Γ be geometrically finite and w0Γ discrete. If either δ>1 or w0 is not externally Γ-parabolic, then (1.1) holds. Remark 1.7. (1) For geometrically finite Γ, if the Riemannian volume of E is finite, then skΓ(w0) < ∞ (Corollary 1.15). (2) It can be proved that if δ ≤ 1andw0 is externally Γ-parabolic, the asymptotic count is of the order T log T if δ = 1 and of the order T if δ<1, instead of Tδ (cf. [25]). (3) When Q(w0) < 0, the orbital counting with respect to the hyperbolic metric balls was obtained by Lax and Phillips [19] for Γ geometrically finite with δ> (n − 1)/2, by Lalley [18] for convex cocompact subgroups and by Roblin [31] for all groups with finite Bowen-Margulis-Sullivan measure. (4) When Q(w0) = 0 and Γ is geometrically finite with δ>(n − 1)/2, a version of Theorem 1.2 was obtained in [17]. 1.4. Equidistribution of expanding submanifolds. In this section, we will describe the main ergodic theoretic ingredients used in the proof of Theorem 1.2. Let E˜ ⊂ T1(Hn)beoneofthefollowing: (1) an unstable horosphere over a horosphere S˜ in Hn; (2) the unit normal bundle of a complete proper connected totally geodesic subspace S˜ of Hn;thatis,S˜ is an isometric imbedding of Hk in Hn for some 0 ≤ k ≤ n − 1. Let Γ be a discrete subgroup of G,andsetE := p(E˜) for the projection p : T1(Hn) → T1(Γ\Hn). n Recall that {νx : x ∈ H } denotes a Patterson-Sullivan density of dimension δ. n Let {mx : x ∈ H } denote a G-invariant conformal density of dimension (n − 1). We consider the following locally finite Borel measures on E˜: − Leb (n 1)βv+ (o,π(v)) + PS δβv+ (o,π(v)) + dμE˜ (v)=e dmo(v ),dμE˜ (v)=e dνo(v ), where o ∈ Hn.NotethatμLeb is the measure associated to the Riemannian volume E˜ form on E˜. The measures μPS and μLeb are invariant under Γ = {γ ∈ Γ:γ(E˜)=E˜} and E˜ E˜ E˜ \ ˜ Leb PS hence induce measures on ΓE˜ E. WedenotebyμE and μE , respectively, the \ ˜ → projections of these measures on E via the projection map ΓE˜ E E induced by p. Let mBR denote the Burger-Roblin measure on T1(Γ\Hn) associated to the conformal densities {νx} in the backward direction and {mx} in the forward direction ([6], [31], and see (3.3)). Let {Gt} denote the geodesic flow on T1(Hn).

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| BMS| ∞ | PS| ∞ ⊂ Theorem 1.8. Suppose that m < and μE < .LetF E be a Borel PS 1 n subset with μ (∂F)=0. For any ψ ∈ Cc(T (Γ\H )), E μPS(F ) (1.2) lim e(n−1−δ)t · ψ(Gt(v)) dμLeb(v)= E · mBR(ψ). → ∞ E | BMS| t + F m In particular, this holds for F = E. See Theorem 3.6 for a version of Theorem 1.8 without the finiteness assumption | PS| on μE . Leb ∞ Remark 1.9. Theorem 1.8 applies to F with μE (F )= as well, provided | PS| ∞ μE < . The proof for this generality requires greater care since it cannot be deduced from the cases of F bounded. It is precisely this general nature of our equidistribution theorem which enabled us to state Theorem 1.2 for general groups | PS| Γ, only assuming the finiteness of the skinning size skΓ(w0)= μE for a suitable E. When E is a horosphere and F is bounded, Theorem 1.8 was obtained earlier by Roblin [31, p. 52]. We were motivated to formulate and prove the result from an independent view point; our attention was especially on the case of π(E)beinga totally geodesic immersion. This case involves many new features, observations, and applications (cf. [23], [24]). The main key to our proof is the transversality theorem, Theorem 3.5, which was influenced by the work of Schapira [34]. The transversality theorem provides a precise relation between the transversal intersections of geodesic evolution of F with a given piece, say T , of a weak stable leaf and the transversal measure corresponding to the mBMS measure on T . For Γ Zariski dense, we generalize Theorem 1.8 to ψ ∈ Cc(Γ\G). To state the n generalization, we fix o ∈ H and X0 ∈ E˜ based at o. Then, for K = Go and Hn 1 Hn M = GX0 ,wemayidentify and T ( ) with G/K and G/M, respectively. Let A = {ar} be the one-parameter subgroup such that the right translation action by r BR ar on G/M corresponds to G .Let¯m denote the measure on Γ\G which is the M-invariant extension of mBR via the natural projection map Γ\G → Γ\G/M = 1 \Hn \ T (Γ ). Let H = GE˜ , and let dh denote the invariant measure on ΓH H whose Leb projection to E coincides with μE . Theorem 1.10. Let Γ be a Zariski dense discrete subgroup of G such that |mBMS| < PS ∞ and |μ | < ∞. Then for any ψ ∈ Cc(Γ\G), E | PS| (n−1−δ)r μE BR lim e ψ(Γhar) dh = m¯ (ψ). r→∞ | BMS| h∈ΓH \H m When Γ is a lattice in G and E is of finite Riemannian volume, Theorem 1.10 is due to Sarnak [33] for horocycles in H2, Randol [28] for unit normal vectors based at a point in the cocompact lattice case in H2, and Duke-Rudnick-Sarnak [9] and Eskin-McMullen [10] in general (also see [15, Appendix]). In §7, we deduce Theorem 1.2 from Theorem 1.8. The standard techniques of orbital counting via equidistribution results require significant modifications due to the fact that mBR is not G-invariant.

PS 1.5. On finiteness of μE for geometrically finite Γ. An important condition PS for the application of Theorem 1.8 is to determine when μE is finite. In this subsection we assume that Γ is geometrically finite. Letting E˜ and S˜ = π(E˜)be

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§ \ ˜ → \Hn as in 1.4, suppose further that the natural imbedding ΓS˜ S Γ is proper;in ˜ \Hn { ∈ ˜ ˜} particular, p(S)isclosedinΓ ,whereΓS˜ = γ Γ:γS = S . ˜ PS When S is a point or a horosphere, μE is compactly supported (Theorem 4.9). ˜ Theorem 1.11 (Theorem 4.7). If S is totally geodesic, then ΓS˜ is geometrically ﬁnite.

Definition 1.12 (Parabolic-corank). Let Λp(Γ) denote the set of parabolic fixed n points of Γ in ∂H . For any ξ ∈ Λp(Γ), Γξ is a virtually free abelian group of rank at least one. Define − ∩ pb-corank(ΓS˜)= max (rank(Γξ) rank(Γξ ΓS˜)) . ξ∈Λp(Γ)∩∂(S˜) ∩ ˜ ∅ If Λp(Γ) ∂(S)= , we set pb-corank(ΓS˜) = 0. In particular, the parabolic corank of ΓS˜ is always zero when Γ is convex cocompact. Lemma 1.13 (Lemma 6.2). If S˜ is totally geodesic, then ≤ ˜ pb-corank(ΓS˜) codim(S). Theorem 1.14 (Theorems 6.3 and 6.4). We have: PS (1) supp(μE ) is compact if and only if pb-corank(ΓS˜)=0. | PS| ∞ (2) μE < if and only if pb-corank(ΓS˜) <δ. 1 ∈ Note that by [8, Prop. 2], δ>2 maxξ Λp(Γ) rank(Γξ). As a consequence of Theorem 1.14, we get: ˜ ≥ | Leb| ∞ Corollary 1.15 (Theorem 6.5). Suppose that dim(S) (n +1)/2.If μE < , | PS| ∞ then μE < . PS Leb ˜ § 1.6. Finiteness of μE or μE and closedness of E. Let E and E be as in 1.4. | Leb| ∞ 1 \Hn In [29], it is shown that μE < implies that E is a closed subset of T (Γ ). PS We prove an analogous statement for μE . Theorem 1.16 (Theorem 2.21). Let Γ be a discrete Zariski dense subgroup of G. | PS| ∞ \ ˜ → \Hn If μE < , then the natural embedding ΓS˜ S Γ is proper.

1.7. Integrability of φ0 and a characterization of a lattice. Deﬁne φ0 ∈ C(Γ\Hn)by n φ0(x):=|νx| for x ∈ Γ\H .

The function φ0 is an eigenfunction of the hyperbolic Laplace operator with eigen- − − − n−1 ∈ value δ(n 1 δ) (see [36]). Sullivan [37] showed that if δ> 2 ,thenφ0 2 n BMS L (Γ\H ,dVolRiem) if and only if |m | < ∞. The following theorem, which is a novel application of Ratner’s theorem [30], relates the integrability of φ0 with the n ﬁniteness of VolRiem(Γ\H ): Theorem 1.17 (§3.6). For any discrete subgroup Γ, the following statements are equivalent: 1 n (1) φ0 ∈ L (Γ\H ,dVolRiem); (2) |mBR| < ∞; (3) Γ is a lattice in G. Although mBR depends on the choice of the base point o, its ﬁniteness is independent of the choice. If Γ is a lattice, then δ = n − 1, and hence φ0 is a constant function by the uniqueness of the harmonic function [38].

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2. Transverse measures 2.1. Let (Hn,d) denote the hyperbolic n-space and ∂Hn its geometric boundary. Let G denote the identity component of the isometry group of Hn.Wedenote by T1(Hn) the unit tangent bundle of Hn and by π the natural projection from T1(Hn) → Hn. By abuse of notation, we use d to denote a left G-invariant metric 1 n on T (H ) such that d(π(u),π(v)) = min{d(u1,v1):π(u1)=π(u),π(v1)=π(v)}. 1 n n n For a subset A of T (H )orH or ∂H and a subgroup H of G,wedenotebyHA the stabilizer subgroup {g ∈ H : g(A)=A} of A in H. Denote by {Gr : r ∈ R} the geodesic ﬂow on T1(H). For u ∈ T1(Hn), we set

(2.1) u+ := lim Gr(u)andu− := lim Gr(u), r→∞ r→−∞

which are the endpoints in ∂Hn of the geodesic deﬁned by u.Notethat(g(u))± = g(u±)forg ∈ G.ThemapViz:T1(Hn) → ∂Hn given by Viz(u)=u+ is called the visual map.

2.2. The Busemann function β : ∂Hn × Hn × Hn → R is deﬁned as follows: for ξ ∈ ∂Hn and x, y ∈ Hn,

(2.2) βξ(x, y) = lim d(x, ξr) − d(y, ξr), r→∞

where ξr is any geodesic ray tending to ξ as r →∞, and the limiting value is independent of the choice of the ray ξr. Note that β is diﬀerentiable and invariant under isometries; that is, for g ∈ G n and x, y ∈ H , βξ(x, y)=βg(ξ)(g(x),g(y)). For u ∈ T1(Hn), the unstable horosphere based at u− is the set

+ 1 n − − H { ∈ H − } u = v T ( ):v = u ,βu (π(u),π(v)) = 0 , and the stable horosphere based at u+ is the set H− { ∈ 1 Hn + + } u = v T ( ):v = u ,βu+ (π(u),π(v)) = 0 . The weak stable manifold corresponding to u is

˜ s −1 + { ∈ 1 Hn + +} Wu =Viz (u )= v T ( ):v = u ,

∈H+ ∈ R ⇒ Gr Gr r (2.3) v1,v2 u , r d( (v1), (v2)) = e d(v1,v2), ∈ ˜ s ≥ ⇒ Gr Gr ≤ (2.4) v1,v2 Wu , r 0 d( (v1), (v2)) d(v1,v2).

The image under π of a stable or an unstable horosphere H in T1(Hn) based at n n ξ is called a horosphere in H based at ξ. Hence π(H)={y ∈ H : βξ(x, y)=0} for x ∈ π(H).

2.3. Let S˜ be one of the following: a horosphere or a complete connected totally geodesic submanifold of Hn of dimension k for 0 ≤ k ≤ n − 1. Let E˜ ⊂ T1(Hn) denote the unstable horosphere with π(E˜)=S˜ if S˜ is a horosphere, and the unit normal bundle over S˜ if S˜ is totally geodesic.

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Lemma 2.1. The visual map Viz restricted to E˜ is a diffeomorphism onto ∂(Hn) ∂(S˜). Proof. The conclusion is obvious if S˜ is a point or a horosphere. Now suppose that S˜ is a totally geodesic subspace of dimension 1 ≤ k ≤ n − 1. Consider the upper half space model for Hn: (2.5) Hn = {x + jy : x ∈ Rn−1,y>0,j=(0,...,0, 1)}, ∼ − and ∂Hn = Rn 1 ∪{∞}. Without loss of generality, we may assume that ∞∈∂(S˜) and hence ∂S˜ {∞} is a (k − 1)-dimensional affine subspace, say F ,ofRn−1. n−1 For any x ∈ R L,letx1 be the orthogonal projection of x on L.Letx2 = n 1 n x1 + x − x1·j ∈ H .Letv ∈ T (H ) be the unit vector based at x2 in the same + direction as x − x1.Thenv ∈ E˜ and v = x. Now the conclusion of the lemma is straightforward to deduce. ˜ H+ ∈ 1 Hn − 2.3.1. Maps between E, and v . For v T ( ), v is the vector with the same ∈ ˜ H+ −1 ˜ → base point as v but in the opposite direction. For v E,letξv : v Viz (∂S) E˜ {−v} be the map given by −1 + (2.6) ξv(u)=Viz (u ) ∩ E.˜ ˜ {− }→H+ −1 ˜ Then ξv is a diffeomorphism. Its inverse, qv : E v v Viz (∂S), is the map given by −1 + ∩H+ (2.7) qv(w)=Viz (w ) v .

Proposition 2.2. There exist C1 > 0 and 0 > 0 such that:

(1) if v, w ∈ E˜ and d(v, w) <0,then

|βw+ (π(qv(w)),π(w))|≤d(qv(w),w)

|βw+ (π(ξv(w)),π(w))|≤d(ξv(w),w) 0 ≤ and C1 > 0 such that Df(u) C1 for any u with d(v, u) <0. Therefore, since f(v)=0,thereexistsC1 > 0 such that |f(u)| = |f(u) − f(v)|≤C1 · d(v, u) for all u ∈ E˜ with d(u, v) <0. This proves (1), and (2) can be proved similarly. Remark 2.3. The following stronger form of statements in Proposition 2.2 hold: There exist 0 > 0 and C1 > 0 such that 2 ˜ |βw+ (π(qv(w)),π(w))|≤C1d(v, w) , for all w ∈ E with d(w, v) <0; | |≤ 2 ∈H+ βw+ (π(ξv(w),π(w))) C1d(v, w) , for all w v with d(w, v) <0. We omit a proof, as the stronger version will not be used in this article. Notation 2.4. Let Γ be a non-elementary torsion-free discrete subgroup of G and set X := Γ\Hn. Both the natural projection maps Hn → X and T1(Hn) → T1(X) will be denoted by p.

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2.4. Boxes, plaques and transversals. Let u ∈ T1(Hn). Consider a relatively H+ compact open set P containing u in u , and a relatively compact open neighbor- −1 + ∈ ∈ H+ hood T of u in Viz (u ). For each t T and p P , the horosphere t intersects Viz−1(p+) at a unique vector: we deﬁne H+ ∩ −1 + ∈ 1 Hn tp := t Viz (p ) T ( ). The map (t, p) → tp provides a local chart of a neighborhood of u in T1(Hn). Since u ∈ P , in this notation tu = t.Wecalltheset B(u)={tp ∈ T1(Hn):t ∈ T,p ∈ P } a box around u if some neighborhood of B(u) injects into T1(X) under p.Wewrite B = B(u)=TP. Note that P (resp. T ) may be disconnected and of ‘large’ diameter, and then the corresponding T (resp. P ) will be chosen to be of small diameter in order to achieve the required injectivity of p on a neighborhood of B(u). For any t ∈ T ,theset { ∈ }⊂H+ tP := tp : p P t is called a plaque at t,andforanyp ∈ P ,theset Tp := {tp : t ∈ T }⊂Viz−1(p+) is called a transversal at p.Theholonomy map between the transversals Tp and Tp is given by tp → tp for all t ∈ T . ∈ ⊂H+ ⊂ −1 + Remark 2.5. If v = tp B,thentP v , Tp Viz (v )andB(v)=(Tp)(tP )is a box about v and TP =(Tp)(tP ). Also, B(u)andB(v) have the same collections of plaques and transversals. For small >0, let −1 + T + = {s ∈ Viz (u ):d(s, T ) <},

T − = {t ∈ T : d(t, ∂T ) >}, and B ± = T ±P. ∈ ⊂H+ ⊂ −1 + Note that for any γ G, γP γu, γT Viz ((γu) ), γ(tp)=(γt)(γp)for any (t, p) ∈ T × P , γ(TP)=γ(B(u)) = B(γu)=(γT)(γP), γ(tP ) is a plaque at γt and γ(Tp) is a transversal at γp.Also,γB ± =(γT)±(γP). For r ∈ R, Gr(B(u)) = B(Gr(u)) = (Gr(T ))(Gr(P )). 2.5. For the rest of this section, let B = TP ⊂ T1(Hn) denote a box such that 1 B 0+ injects into T (X)forsome0 > 0. By choosing a smaller 0 if necessary, let C1 > 0 be such that Proposition 2.2 holds. Let { ∈ ∈ } (2.8) C2 =max d(tp1,tp2):t T 0+,p1,p2 P . In this section we will develop auxiliary results to understand the intersection of Gr(E) with p(B)forr 1. First we will show that for any γ ∈ ΓifGr(γE˜) ∩ B ∈ ∩Gr ˜ Gr ˜ is non-empty, there exists a unique t T 0+ (γE)andthesets (γE)and r −r G (tP )arecontainedinC1C2e -tubular neighborhoods of each other. Lemma 2.6. Let r ∈ R and γ ∈ Γ. Suppose that Gr(γE˜) tp for some t ∈ T , ∈ G−r −1 ∈ ˜ ∈ G−r −1 ∈H+ p P .Letv = (γ tp) E.Letp1 P , y = (γ tp1) v ,and + + w = ξv(y) ∈ E˜.Thenw = y , −r −r d(v, y) ≤ C2e and d(y, w) ≤ C1C2e .

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+ + ∈H+ Proof. By (2.6), w = y .Sincetp, tp1 t , by (2.3) and (2.8), −r −1 −r −1 −r −r d(v, y)=d(G (γ tp), G (γ tp1)) = d(G (tp), G (tp1)) −r −r ≤ d(tp, tp1)e ≤ C2e . −r By Proposition 2.2, d(y, w)=d(y, ξv(y)) ≤ C1d(v, y) ≤ C1C2e . Lemma 2.7. For any r ∈ R and γ ∈ Γ, # T ∩Gr(γE˜) =#(G−r(γ−1T ) ∩ E˜) ≤ 1. Proof. Since Viz(G−r(γ−1T )) = γ−1 Viz(T ) is a singleton set and Viz restricted to E˜ is injective, the conclusion follows. Notation 2.8. For r ∈ R and γ ∈ Γ, in view of Lemma 2.7, deﬁne −r −1 ˜ r ˜ ξG−r(γ−1t)(G (γ tP )) ⊂ E if T ∩G (γE)={t}, (2.9) E˜r,γ = ∅ if T ∩Gr(γE˜)=∅.

Proposition 2.9. For any 0 <≤ 0, r>r := log(C1(C1 +1)C2/) and γ ∈ Γ, we have −r −1 −r −1 (2.10) G (γ B −) ∩ E˜ ⊂ E˜r,γ ⊂G (γ B +) ∩ E.˜

Proof of ﬁrst inclusion in (2.10). Let γ ∈ Γ, t ∈ T − and p ∈ P be such that −r −1 −r −1 v := G (γ tp) ∈ E˜.Lety = G (γ t)andw = ξv(y) ∈ E˜. By Lemma 2.6, −r d(y, w) ≤ C1C2e </(C1 +1)<. Gr Gr + + + + Let t1 = (γw). Since t = (γy)andw = y , t1 = t . By (2.4), r r d(t, t1)=d(G (γy), G (γw)) ≤ d(γy,γw)=d(y, w) <. + + Therefore t1 ∈ T ,fort ∈ T −. Since (t1p) =(tp) ,wehave −r −1 + −r −1 + + G (γ t1p) = G (γ tp) = v . G−r −1 G−r −1 ∈H+ ∈ ˜ Since w = (γ t1), (γ t1p) w .Also,w, v E. Therefore by (2.6), −r −1 v = ξw(G (γ t1p)) ∈ E˜r,γ .

Proof of second inclusion in (2.10). By Lemma 2.7, let {t} = T ∩Gr(γE˜)forsome −r −1 −r −1 γ ∈ Γ. Let v = G (γ t) ∈ E˜, p ∈ P , y = G (γ tp), and w = ξv(y) ∈ E˜r,γ .By Lemma 2.6, −r −r d(v, w) ≤ d(v, y)+d(y, w) ≤ C2e + C1C2e ≤ /C1. ∈H+ + + Put v1 = qw(v) w . By (2.7), v1 = v , and by Proposition 2.2(1),

d(v, v1) ≤ C1d(v, w) ≤ . Gr ∈H+ Gr + + Put t1 = (γv1) Gr (γw).Sincet = (γv), t1 = t .By(2.4), r r d(t, t1)=d(G (γv), G (γv1)) ≤ d(γv,γv1) ≤ . ∈ Gr ∈H+ + + Hence t1 T +.Now (γw),t1p t1 .Sincew = y , r + r + + + (G (γw)) =(G (γy)) =(tp) =(t1p) . H+ Gr ∈G−r −1 Since Viz is injective on t1 , (γw)=t1p. Hence w (γ B +).

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2.6. Measure on E corresponding to a conformal density on ∂Hn. Let n n {μx : x ∈ H } be a Γ-invariant conformal density of dimension δμ > 0on∂H . n n That is, for each x ∈ H , μx is a positive ﬁnite Borel measure on ∂H such that for all y ∈ Hn, ξ ∈ ∂Hn and γ ∈ Γ, dμ x δμβξ(y,x) (2.11) γ∗μx = μγx and (ξ)=e , dμy −1 n where γ∗μx(F ):=μx(γ (F )) for any Borel subset F of ∂H . Fix o ∈ Hn. We consider the measure on E˜ given by

δμβv+ (o,π(v)) + (2.12) dμE˜ (v)=e dμo(v ). ∈ Hn By (2.11), μE˜ is independent of the choice of o and γ∗μE˜ = μγE˜ for any ∈ \ ˜ γ Γ. Let μΓ \E˜ be the locally finite Borel measure on ΓE˜ E induced by μE˜ as E˜ ∈ ˜ ¯ ∈ ˜ → ¯ follows: For any f Cc(E), let f(ΓE˜ v)= γ∈Γ f(γv), for all v E.Thenf f is a surjective map from Cc(E˜)fromtoCc(Γ ˜ \E˜), and E ¯ (2.13) fdμΓ \E˜ := fdμE˜ \ ˜ E˜ ˜ ΓE˜ E E is well defined; see [29, Chapter 1] for a similar construction. Now let μE be the measure on E = p(E˜) defined as the pushforward of μ \ ˜ ΓE˜ E \ ˜ 1 \Hn → ⊂ 1 Hn from ΓE˜ E to T (Γ ) under the map ΓE˜ v Γv. Thus for any set B T ( ) such that p is injective on B and for any measurable non-negative function f on E ∩ p(B), ∩ fdμE = [γ]∈Γ/Γ ∈ ˜∩ f(p(u)) dμγE˜ (u) (2.14) E p(B) E˜ u γE B = ∈ ∈ ˜∩ −1 f(p(u)) dμ ˜ (u), [γ] Γ/ΓE˜ u E γ B E where the integration over an empty set is defined to be 0. Therefore by Proposi- tion 2.9 we obtain the following:

Proposition 2.10. Let 0 <≤ 0 and r>r . Then for all Borel measurable 1 functions Ψ ≥ 0 on T (X) with supp(Ψ) ⊂ p(B −) and f ≥ 0 on E, we have Gr Gr Ψ( (u))f(u) dμE (u)= −r Ψ( (u))f(u) dμE(u) u∈E E∩p(G (B±)) Gr = ∈ G−r −1 ∩ ˜ Ψ( (u))f(u) dμ ˜ (u) [γ] Γ/ΓE˜ (γ B±) E E r = ∈ ˜ Ψ(G (p(u)))f(p(u)) dμ ˜ (u). [γ] Γ/ΓE˜ Er,γ E Remark 2.11. (1) For the counting application in §7, we will use the results of this \ ˜ → 1 \Hn section only for the case when the map ΓE˜ E T (Γ ) is proper, in which case μE is a locally finite Borel measure. (2) In the general case, μE may not be σ-finite, but it is an s-finite measure; namely, a countable sum of finite measures (with possibly non-disjoint supports). ˜ ˜ Hn − \ ˜ 1 \Hn (3) If the dimension of S = π(E)in is 0 or n 1, the map ΓE˜ E to T (Γ ) 1 n is injective, and hence μE is σ-finite on T (Γ\H ). 2.6.1. Measures on horospherical foliation and their semi-invariance under geodesic flow. The conformal density {μx} induces a Γ-equivariant family of measures 1 n 1 n {μ + : u ∈ T (H )} on the unstable horospherical foliation on T (H ): Hu

δμβ + (o,π(v)) + (2.15) dμ + (v)=e v dμ (v ). Hu o

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For any r ∈ R,sinceGr(v)+ = v+ and r r βv+ (o, π(G (v))) − βv+ (o, π(v)) = βv+ (π(v),π(G (v))) = r, by (2.11), we get for all γ ∈ Γandr ∈ R,

r −δμr (2.16) γ∗μH+ = μH+ and G∗μH+ = e μH+ . u γu u Gr (u)

r 2.7. On transversal intersections of, G (ΓE˜) with B. Let a box B, 0 > 0, C1 > 0andC2 > 0 be as described in the beginning of §2.5. For any 0 <≤ 0, we put

(2.17) r =log((C1 +1)C1C2/).

r Proposition 2.12. Let 0 <≤ 0, r>r ,and{t} = T ∩G (γE˜) for some γ ∈ Γ. Then for all measurable functions Ψ ≥ 0 on B and f ≥ 0 on E˜, 0+ − − − − − δμ G r 1 (e )f ( (γ t)) Ψ dμH+ tP t ≤ δμr Gr e Ψ( (γw))f(w)dμE˜ (w) ∈ ˜ w Er,γ − − ≤ δμ + G r 1 + (e )f ( (γ t)) Ψ dμH+ , tP t ± ˜ ± where f on E and Ψ on B + are deﬁned as f +(u)=sup f(u ), {u1∈E˜:d(u1,u)≤ } 1 f −(u) = inf f(u ), {u1∈E˜:d(u1,u)≤ } 1 (2.18) + Ψ (tp)=sup{t1∈T+:d(t1p,tp)≤ } Ψ(t1p), − Ψ (tp) = inf{t1∈T+:d(t1p,tp)≤ } Ψ(t1p). G−r −1 ∈ ˜ ⊂H+ → ˜ ⊂ ˜ Proof. Let v = (γ t) E.Letφ : tP t Er,γ E be the map given by −r −1 φ(tp)=w := ξv(y), where p ∈ P and y = G (γ tp). By Lemma 2.6, −r −r (2.19) d(y, w)

δμr (2.22) e dμH+ (y)=dμH+ (tp). v t + + For the map y → w = ξv(y), by (2.12) and (2.15), since w = y , δ β (o,π(w)) e μ w+ δμβ + (π(y),π(w)) (2.23) dμ ˜ (w)= dμ + (y)=e w dμ + (y). E δ β (o,π(y)) Hv Hv e μ y+

By (2.19), |βw+ (π(y),π(w))|≤d(π(y),π(w)) ≤ . Therefore,

−δμ δμ (2.24) e

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Combining (2.22) and (2.24), for the map w = φ(tp)weget

− dμ ˜ (w) (2.25) e δμ ≤ eδμr E ≤ eδμ . dμH+ (tp) t By noting that G−r(γ−1t)=v and tp = Gr(γy), the conclusion of the proposition follows from (2.20), (2.21) and (2.25).

Notation 2.13. For r ≥ 0andt ∈ T ∩Gr(ΓE˜), in view of Lemma 2.7 let ¯ { ∈ { } ∩Gr ˜ } (2.26) Γr,t = [γ] Γ/ΓE˜ : t = T (γE) . ∈ Since p is injective on B 0+, for notational convenience we identify t T 0+ with its image p(t) ∈ p(T ) ⊂ X. Therefore we have { ∈ ˜ ∅} ¯ ¯ (2.27) [γ] Γ/ΓE˜ : Er,γ = = Γr,t = Γr,t. t∈T ∩Gr (ΓE˜) t∈T ∩Gr (E) Combining Proposition 2.10 and Proposition 2.12, in view of (2.27) we deduce the following:

Corollary 2.14. Let 0 <≤ 0 and r>r . For all measurable functions Ψ ≥ 0 ⊂ ≥ on B 0+ with supp(Ψ) B − and f 0 on E, we have − − − − δμ ¯ r (e ) #(Γr,t)f (G (t)) · Ψ dμH+ t ∈ ∩Gr tP t T (E)

δμr r ≤ e Ψ(G (u))f(u) dμE (u) E − δμ ¯ + r + ≤ (e ) #(Γr,t) · f (G (t)) · Ψ dμH+ , t t∈T ∩Gr (E) tP ± ± where f on B + and Ψ on E are deﬁned as in (2.18). 2.8. Haar system and admissible boxes. Lemma 2.15 ([31]). For a uniformly continuous Ψ ∈ C(B),themap

t ∈ T → Ψ dμH+ tP t

is uniformly continuous. In particular, the map t → μH+ (tP ) is uniformly contin- t uous. Proof. Note that (tp)+ = p+. Therefore by (2.15)

δμβ + (o,π(tp)) + Ψ dμH+ = Ψ(tp)e p dμo(p ). tP t P δ β (o,π(tp)) Put φ(tp)=Ψ(tp)e μ p+ .Sinceφ is uniformly continuous on B, Ψ dμH+ − Ψ dμH+ ≤ μo(Viz(P )) · sup|φ(t1p) − φ(t2p)|→0 t1 t2 ∈ t1P t2P p P

as d(t1,t2) → 0.

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Definition 2.16. AboxB = TP as defined in §2.4 is called admissible with respect to the conformal density {μx} if every plaque of B has a positive measure with respect to {μH+ };thatis,μH+ (tP ) > 0 for all t ∈ T or, equivalently, t + n μx(Viz(tP )) = μx(P ) > 0 for some (and hence all) x ∈ H . n Lemma 2.17. Fix a conformal density {μx}x∈Hn on ∂H . Then for any u ∈ 1 n T (H ), there exists an admissible box around u with respect to {μx}. n Proof. Fix any x ∈ H . Since Γu− is virtually abelian and since we assume that Γ is non-elementary, Γ does not fix u−. Therefore by the Γ-invariance and the { } { −} H+ → conformality of the density μx , we have supp(μx) = u .SinceViz: u Hn { −} ∈H+ + ∈ ∂ u is a diffeomorphism, there exists u1 u such that u1 = Viz(u1) supp(μ ). If γu = u for any γ ∈ Γ, then by the conformality, u ∈ supp(μ + )and x 1 Hu we replace u1 by u.Sincep is injective on {u, u1}, there exists a relatively compact H+ { } open subset P of u containing u, u1 such that p is injective on an open set Ω 1 n of T (H ) containing P .Thenμx(Viz(P )) > 0. By Lemma 2.15, we can choose T a large enough ball in Viz−1(u+) so that some neighborhood of the closure of B = TP iscontainedinΩ.NowB = TP is an admissible box.

2.8.1. Let B = TP be an admissible box with respect to a conformal density {μx}

such that p is injective on a neighborhood of the closure of B 0+ for some 0 > 0. Let C1, C2 be as described at the beginning of §2.5. For notational convenience, 1 we will identify T 0+ and B 0+ with their respective images in T (X) under p.

Proposition 2.18. Let 0 <≤ 0 and r>r (see (2.17)). Then for all measurable functions ψ ≥ 0 on T with supp(ψ) ⊂ T − and f ≥ 0 on E, we have 0+ − − − δμ Gr (e ) Ψ ( (w))f (w) dμE(w) E −δμr −r ≤ e #(Γ¯r,t) · ψ(t)f(G (t)) ∈ ∩Gr t T (E) − ≤ δμ + Gr + (e ) Ψ ( (w))f (w)dμE(w), E

where the function Ψ on B 0+ is deﬁned by

+ ∈ × Ψ(p(tp)) := ψ(t)/μH (tP ), for all (t, p) T 0+ P, t and Ψ± on B and f ± on E are deﬁned as in (2.18). + Proof. Since Ψ dμH+ = ψ(t), the result is straightforward to deduce from Corol- tP t lary 2.14. In §3, Proposition 2.18 will enable us to describe the limiting distribution of the transversal intersections T ∩Gr(E) using the mixing of the geodesic ﬂow with respect to mBMS (cf. Theorem 3.5). 2.9. Some direct consequences. The results proved in this subsection are also of independent interest. Let the notation be as in §2.8.1.

Corollary 2.19. Let 0 <≤ 0 and f be a measurable function on E such that f + ∈ L1(E,μ ). Then for any r>r and any measurable function ψ on T , E −r #(Γ¯r,t) ·|ψ(t)f(G (t))| < ∞. t∈T ∩Gr (E)

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In particular, if there exists a Γ-invariant conformal density {μx} and |μE | < ∞, then #(Γ¯r,t) < ∞. t∈T ∩Gr E

Proof. By Proposition 2.18 with T in place of T and declaring ψ to be zero outside T , we obtain the first claim because − ¯ ·| G r |≤ δμr + · | +| #(Γr,t) ψ(t)f( (t)) (1 + )e Ψ ∞ μE( f ). t∈T ∩Gr (E) To deduce the second claim from the first one, we choose f =1onE and ψ =1on T . Definition 2.20 (Radial limit points). The limit set Λ(Γ) of Γ is the set of all n accumulation points of an orbit Γ(z)inH for z ∈ Hn. As Γ acts properly discontinuously on Hn, Λ(Γ) is contained in ∂Hn. Apointξ ∈ Λ(Γ) is called a radial limit point if for some (and hence every) geodesic ray β tending to ξ and some (and hence every) point x ∈ Hn, there is a sequence γi ∈ Γ with γix → ξ,andd(γix, β) is bounded. We denote by Λr(Γ) the set of radial limit points for Γ. If Γ is non-elementary, Λr(Γ) is a non-empty Γ-invariant subset of Λ(Γ). Since n Λ(Γ) is a Γ-minimal closed subset of ∂H ,wehavethatΛr(Γ) = Λ(Γ). Theorem 2.21. Let C denote the smallest subsphere of Hn containing Λ(Γ). Sup- pose that C = ∂S˜ or dim(C) > dim(∂S˜).IfthereexistsaΓ-invariant conformal { ∈ Hn} | | ∞ \ ˜ → density μx : x such that μE < , then the natural map p¯ :ΓE˜ E Γ\ T1(Hn) is proper.

Proof. Note that Γ ⊂ GC = {g ∈ G : gC = C}, because if γ ∈ Γ, then γC ∩ C ⊃ Λ(Γ), hence by minimality γC = C. ˜ ∩ Suppose C = ∂S. Then, since GS˜ = G∂S˜,Γ=Γ GC =ΓS˜ =ΓE˜ , and hence the properness of p¯ is obvious. Now suppose that dim(C) > dim(S˜) and that that p¯ is not proper. Then there 1 n exist sequences γi ∈ Γandei ∈ E˜ such that γiei converges to a vector v ∈ T (H ) as i →∞,and

(2.28) γiΓE˜ = γj ΓE˜ , for all i = j. ∈ ˜ ˜ ∈ Fix e0 E.SinceGE˜ acts transitively on E,thereexistshi GE˜ such that ei = hie0.Thenγihie0 converges to v. Therefore there exists g ∈ G such that γihi → g and v = ge0. Now Viz(gE˜)=∂Hn − ∂(gS˜). Since dim(∂(gS˜)) = dim(S˜) < dim(C), we have that Λ(Γ) ∂(gS˜) is a non-empty open subset of Λ(Γ). Since Λr(Γ)isdensein Λ(Γ), it follows that

Λr(Γ) ∩ Viz(gE˜) = ∅. ∈ + ∈ Therefore there exists h0 GE˜ such that Viz(gh0e0)=(gh0e0) Λr(Γ). Hence ri 1 there exist ri →∞such that p¯(G (gh0e0)) converges to a point in T (X). Then { }⊂ Gri → ∈ 1 Hn there exists a sequence γi Γ such that (γigh0e0) u for some u T ( ). Let B = TP be an admissible box centered at u.Let>0 be such that ∈ ∈ N u B3 −.Fixk such that rk >r (see (2.17)) such that for γ = γk,wehave r G (γ gh0e0) ∈ B2 −.

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r Since γihi → g, G (γ γihih0e0) ∈ B − for all i ≥ i0 for some i0.Sincehih0e0 ∈ r E˜, by (2.10) ti ∈ T ∩G (γ γiE˜) for all i ≥ i0. Therefore,

r −1 (2.29) (ΓT ∩G E˜) ⊃{(γ γi) ti : i ≥ i0}. We claim that for any i ∈ N, −1 −1 −1 −1 (2.30) ΓE˜ γi (γ ) ti =ΓE˜ γj (γ ) tj , for all but ﬁnitely many j.

To see this, since p is injective on T ,ifti = tj ,thenΓti =Γ tj , and hence (2.30) ∩ −1 holds. If ti = tj , then it follows from (2.28) as Γ G(γ ) ti is ﬁnite. Combining (2.29) and (2.30), we get that \ ∩Gr ˜ ∞ #(ΓE˜ (ΓT E)) = . ∈ ∩Gr \ ∩Gr ˜ ¯−1 | PS| ∞ We observe that if t T (E), then ΓE˜ (Γt E)=Γr,t t.If μE < ,then by (2.19) of Corollary 2.19 \ ∩Gr ˜ ≤ ¯ ∞ #(ΓE˜ (ΓT E)) #(Γr,t) < , t∈T ∩Gr (E) which is a contradiction.

Remark 2.22. (1) Theorem 2.21 holds for Γ Zariski dense: since Γ ⊂ GC and GC is Zariski closed, we have C = ∂Hn for Γ Zariski dense. ˜ ˜ ⊂ (2) Theorem 2.21 holds in the case Λ(ΓS˜)=∂S,sinceS C in this case, and hence we have that either S˜ = C or dim(C) > dim(S˜).

Equidistribution of Gr Leb 3. ∗μE 3.1. BMS-measure and BR-measure on T1(X). As before, let Γ be a non- n elementary torsion-free discrete subgroup of G and set X := Γ\H .Let{μx} and { } Hn μx be Γ-invariant conformal densities on ∂ of dimension δμ and δμ , respectively. Following Roblin [31], we deﬁne a measure mμ,μ on T1(X) associated to { } { } ∈ Hn μx and μx as follows. Fix o . Then the map + − u → (u ,u ,βu− (o, π(u))) is a homeomorphism between T1(Hn) with (∂Hn × ∂Hn {(ξ,ξ):ξ ∈ ∂Hn}) × R.

Hence we can deﬁne a measurem ˜ μ,μ on T1(Hn)by

− μ,μ δμβu+ (o,π(u)) δμ βu− (o,π(u)) + (3.1) dm˜ (u)=e e dμo(u )dμo(u )ds,

μ,μ where s = βu− (o, π(u)). Note thatm ˜ is Γ-invariant. Hence it induces a locally ﬁnite measure mμ,μ on T1(X) such that if p is injective on Ω ⊂ T1(Hn), then

mμ,μ (p(Ω)) =m ˜ μ,μ (Ω). This deﬁnition is independent of the choice of o ∈ Hn. Two important conformal densities on Hn that we will consider are the Patterson- Sullivan density and the G-invariant (Lebesgue) density.

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3.1.1. Critical exponent δΓ. We denote by δΓ the critical exponent of Γ which is −sd(o,γ(o)) deﬁned as the abscissa of convergence of a Poincar´e series γ∈Γ e for n some o ∈ H ;thatis,theseriesconvergesfors>δΓ and diverges for s<δΓ,and the convergence property is independent of the choice of o ∈ Hn. As Γ is non-elementary, we have δΓ > 0. Generalizing the work of Patterson [26] n for n = 2, Sullivan [36] constructed a Γ-invariant conformal density {νx : x ∈ H } of dimension δΓ supportedonΛ(Γ),whichisuniqueuptohomothetyandiscalled the Patterson-Sullivan density. From now on, we will simply write δ instead of δΓ. n We denote by {mx : x ∈ H } a G-invariant conformal density on the boundary n ∂H of dimension (n−1), which is unique up to homothety, and each mx is invariant under the maximal compact subgroup Gx. It will be called the Lebesgue density. The measure mν,ν on T1(X)iscalledtheBowen-Margulis-Sullivan measure BMS m associated with {νx} ([5], [20], [37]):

BMS δβ + (o,π(u)) δβ − (o,π(u)) + − (3.2) dm (u)=e u · e u dνo(u )dνo(u )ds. The measure mν,m is called the Burger-Roblin measure mBR associated with {νx} and {mx} ([6], [31]):

BR (n−1)β + (o,π(u)) δβ − (o,π(u)) + − (3.3) dm (u)=e u · e u dmo(u )dνo(u )ds. We note that the support of mBMS and mBR are given respectively by {u ∈ T1(X):u+,u− ∈ Λ(Γ)} and {u ∈ T1(X):u− ∈ Λ(Γ)}. 3.2. Relation to classification of measures invariant under horocycles. Burger [6] showed that for a convex cocompact hyperbolic surface Γ\H2 with δ> 1/2, mBR is a unique ergodic horocycle invariant locally finite measure which is not supported on closed horocycles. Roblin extended Burger’s result in much greater n generality. By identifying the space ΩH of all unstable horospheres with ∂H × R + − δs by H (u) → (u ,βu− (o, π(u))), one defines the measure dμˆ(H)=dνo(ξ)e ds for H =(ξ,s). Then Roblin’s theorem [31, Thm. 6.6] says that if |mBMS| < ∞,then μˆ is the unique Radon Γ-invariant measure on Λr(Γ) × R ⊂ ΩH.Thisimportant classification result is not used in this article, but it suggests that the asymptotic distribution of expanding horospheres should be described by mBR. ˜ H+ ˜ 3.3. Patterson-Sullivan and Lebesgue measures on E, u and E. Let S and E˜ be as in §2.3. The following measures are special cases of the measures defined in §2.6. Fix o ∈ Hn. Define the Borel measure μLeb on E˜ such that E˜ − Leb (n 1)βv+ (o,π(v)) + (3.4) dμE˜ (v)=e dmo(v ). Since {m } is a G-invariant conformal density on ∂Hn, the measure μLeb is G- x E˜ Leb Leb invariant; that is, g∗μ = μ . In particular, it is a G invariant measure on E˜ g(E˜) E˜ E˜. Define the Borel measure μPS on E˜ such that E˜

PS δβv+ (o,π(v)) + (3.5) dμE˜ (v)=e dνo(v ). We note that μPS is a Γ-invariant measure. E˜ § Leb PS ˜ As described in 2.6, we denote by μE and μE the measures on E = p(E) induced by μLeb and μPS, respectively. Each of them is a pushforward of the E˜ E˜ \ ˜ corresponding locally ﬁnite measure on ΓE˜ E.

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§ PS { PS } Leb { Leb} As in 2.6.1, we have families of measures μ = μH+ and μ = μH+ on the unstable horospherical foliation satisfying PS Gr δr PS Leb Gr (n−1)r Leb μGr (H+)( (F )) = e μH+ (F )and μGr (H+)( (F )) = e μH+ (F ) for any Borel subset F of p(H+).

3.4. Transverse measures for mBMS. For each measurable T contained in a weak stable leaf of the geodesic ﬂow on T1(Hn), called a transversal, deﬁne a measure λT on T by −δs − (3.6) dλT (t)=e dνo(t )ds, − − where s = βt− (o, π(t)). If B = TP is any box and p ∈ P ,then(tp) = t and + + H H − − tp = t , and hence β(tp) (o, π(tp)) = βt (o, π(t)). Hence

dλTp(tp)=dλT (t);

that is, λT is holonomy invariant, where the holonomy is given by t → tp. Now for any Ψ ∈ C(B), by (3.2)-(3.6), we have BMS PS (3.7) fdm = Ψ(tp) dμH+ (tp)dλT (t), B T P t BR Leb (3.8) fdm = Ψ(tp) dμH+ (tp)dλT (t). B T P t 3.4.1. Backward admissible box. Lemma 3.1. For any u ∈ T1(Hn) and >0, there exists a box B = TP about u such that − (1) |λT | > 0, or equivalently νo({t : t ∈ T }) > 0,and Gr Gr ∈ ∈ (2) lim supr→∞ d( (tp), (t p)) <, for all t, t T and p P . Such a box B as above will be called a backward admissible box with asymptotically -thin transversals.

Proof. As in the proof of Lemma 2.17, there exists a relatively compact open − H− { − ∈ −} neighborhood P of u in u such that νo( t : t P ) > 0, and p is in- − − jective on a neighborhood of the closure of P .Letr0 = − log(/4diam(P )). − Then diam(Gr0 (P )) = /4andp is injective on a neighborhood of the closure r0 − r0 − of G (P ). Let T1 be an open relatively compact neighborhood of G (P )in − 1 + Gr0 H+ Viz (u )andP1 be an open relatively compact neighborhood of (u)in Gr0 (u) r0 such that T1P1 is a box about G (u) contained in a ball of radius /2aboutu. −r0 −r0 Let T = G (T1)andP = G (P1). Then B = TP has the required properties. Property (1) holds because

− − − r0 − − − {t : t ∈ T } = {t : t ∈ T1}⊃{t : t ∈G (P )} = {t : t ∈ P }. Gr0 Gr0 Gr0 ∈ For property (2), let t1 = (t)andt1 = (t )inT1 and p1 = (p) P1.Since + + (t1p1) =(t1p1) , for any r>r0, − − Gr Gr Gr r0 Gr r0 ≤ ≤ d( (tp), (t p)) = d( (t1p1), (t1p1)) d(t1p1,t1p1) .

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BMS 3.5. Mixing of the geodesic ﬂow. We assume that |m | < ∞ for the rest of −δd(o,γo) ∞ this section. This implies that Γ is of divergent type; that is, γ∈Γ e = and the Γ-invariant conformal density of dimension δ is unique up to homothety (see [31, Coro. 1.8]). + Hence, up to homothety, νx is the weak-limit as s → δ of the family of measures 1 ν (s):= e−sd(x,γo)δ , x,o e−sd(o,γo) γo γ∈Γ γ∈Γ

n where δγo denotes the unit mass at γo for some o ∈ H . The most crucial ergodic theoretic result involved in this work is the mixing of geodesic ﬂow which was obtained by Rudolph [32] for a geometrically ﬁnite Γ, and by Roblin[31], as well as Babillot [1], in a much greater generality:

2 1 BMS Theorem 3.2 (Rudolph, Roblin, Babillot). For any Ψ1 ∈ L (T (X),m ) and 2 1 BMS Ψ2 ∈ L (T (X),m ), 1 lim Ψ (Gr(x))Ψ (x) dmBMS(x)= mBMS(Ψ ) · mBMS(Ψ ). →∞ 1 2 | BMS| 1 2 r T1(X) m From this theorem, we derive the following result, which generalizes the corresponding result for PS-measures on unstable horospheres due to Roblin [31, Corol- lary 3.2].

∈ 1 ∈ 1 PS Theorem 3.3. For any Ψ Cc(T (X)) and f L (E,μE ), μPS(f) (3.9) lim Ψ(Gr(x))f(x) dμPS(x)= E · mBMS(Ψ). →∞ E | BMS| r x∈E m We will deduce the above statement from its following version.

∈ 1 1 BMS ∈ 1 PS Proposition 3.4. Let Ψ L (T (X),m ) and f L (E,μE ) both be non- negative, bounded and vanish outside compact sets. Then for any >0, PS Gr PS ≤ μE (f) · BMS + (3.10) lim sup Ψ( (x))f(x) dμE (x) m (Ψ ), →∞ |mBMS| r x∈E μPS(f) (3.11) lim inf Ψ(Gr(x))f(x) dμPS(x) ≥ E · mBMS(Ψ−), →∞ E | BMS| r x∈E m

where, for any u ∈ T1(Hn),

+ { ∈ −1 + } Ψ (p(u)) := sup Ψ(p(v)) : d(v, u) <,v Viz (u ) , (3.12) − { ∈ −1 + } Ψ (p(u)) := inf Ψ(p(v)) : d(v, u) <,v Viz (u ) .

Proof. By Lemma 3.1, there exists a ﬁnite open cover B of supp(f) ⊂ E ⊂ T1(X) consisting of backward admissible boxes B with asymptotically -thin transversals; 1 n we identify B ⊂ T (H ) with p(B). By considering a partition of unity subordinate ∈ 1 PS to this cover, f = B∈B φB,whereφB L (E,μE ) is a non-negative function whose support is contained in p(B). Therefore it is enough to prove (3.10) and (3.11) for φB in place of f for each B ∈B.

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∈B ∈ ∈ ˜ Fix any B .Foreach[γ] Γ/ΓE˜ ,letφγ (w)=φB(w) for all w γE.By (2.14), PS PS μE (φB)= μγE˜ (φγ)and ∈ [γ] Γ/ΓE˜ Gr PS Gr PS Ψ( (x))φB(x) dμE (x)= Ψ( (w))φγ (w) dμ ˜ (w). ˜ γE x∈E ∈ w∈γE∩B [γ] Γ/ΓE˜

Therefore to prove (3.10) and (3.11) for φB in place of f, it is enough to prove the following: for any γ ∈ Γandφ := φ ∈ L1(γE,μ˜ PS ) vanishing outside γE˜ ∩ B,we γ γE˜ have PS μ ˜ (φ) (3.13) lim sup Ψ(Gr(w))φ(w) dμPS (w) ≤ γE mBMS(Ψ+), γE˜ | BMS| r→∞ w∈γE˜∩B m PS μ ˜ (φ) (3.14) lim inf Ψ(Gr(w))φ(w) dμPS (w) ≥ γE mBMS(Ψ−). →∞ γE˜ | BMS| r w∈γE˜∩B m Now we express B = TP.IfγE˜ ∩ B = ∅, then both sides of (3.13) are zero, and hence the claim is true. Otherwise, there exists (t1,p1) ∈ T × P such that ∈ ˜ § H+ · −1 ˜ → ˜ {− } v := t1p1 γE. We recall that as in 2.3.1, ξv : v (γ Viz (∂S)) γE v ˜ {− }→H+ · −1 ˜ and qv : γE v v (γ Viz (∂S)) are diﬀerentiable inverses of each other. Letting

P1 = {p ∈ P : ξv(t1p) ∈ Tp}, we claim that

(3.15) γE˜ ∩ B = {ξv(t1p):p ∈ P1}. To see this, if tp ∈ γE˜ for some (t, p) ∈ T × P ,then H+ ∩ −1 + H+ ∩ −1 + qv(tp)= v Viz ((tp) )= t1 Viz (p )=t1p.

Hence ξv(t1p)=tp,andsop ∈ P1. The opposite inclusion is obvious. We deﬁne a map ρ : TP → γE˜ as follows:

ρ(tp)=ξv(t1p), for all (t, p) ∈ T × P. For any t ∈ T , for the restricted map ρ : tP → γE˜, by (2.12) and (2.15), and since (tp)+ = p+ = ρ(tp)+,wehave

PS PS βp+ (π(tp),π(ρ(tp))) (3.16) dμ ˜ (ρ(tp))/dμH+ (tp)=e . γE t In view of this, we deﬁne Φ ∈ L2(T1(X),μBMS) as follows: Φ(x)=0ifx ∈ X B and β (π(tp),π(ρ(tp))) (3.17) Φ(tp)=φ(ρ(tp))e p+ if x = tp ∈ B. We note that

(3.18) Φ(tp) =0 ⇒ ρ(tp) ∈ B ⇒ p ∈ P1. r Also, for t ∈ T and p ∈ P1,wehave{ρ(tp),tp}⊂Tp.SinceG (B)has-thin transversals as r →∞(see Lemma 3.1(2)), r r (3.19) lim sup d(G (ρ(tp)), G (tp)) ≤ for all p ∈ P1. r→∞

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By Theorem 3.2, 1 (3.20) mBMS(Ψ+) · mBMS(Φ) | BMS| m (3.21) = lim Ψ+(Gr(x))Φ(x) dmBMS(x) r→∞ B + Gr PS (3.22) = lim Ψ ( (tp))Φ(tp) dμH+ (tp) dλT (t) r→∞ ∈ ∈ t t T p P1 + Gr PS (3.23) = lim Ψ ( (tp))φ(ρ(tp)) dμγE˜ (ρ(tp)) dλT (t) r→∞ ∈ ∈ t T p P1 ≥| |· Gr PS (3.24) λT lim sup Ψ( (w))φ(w) dμγE˜ (w), r→∞ w∈γE˜∩B where (3.22) follows from (3.7) and (3.18), (3.23) follows from (3.15), (3.16) and (3.17), and to justify (3.24) we put w = ρ(tp) and use (3.12) and (3.19). + By putting Ψ(x)=1=Ψ (x) in (3.21)–(3.24) with equality in (3.24), we get BMS | |· PS ∞ (3.25) m (Φ) = λT μγE˜ (φ) < . Now (3.13) is deduced by comparing (3.20), (3.24) and (3.25), and noting that |λT | = 0 by the backward admissibility of B. Similarly we can deduce (3.14).

Proof of Theorem 3.3. Since both the sides of (3.9) are linear in Ψ, it is enough to prove it for Ψ ≥ 0. Since Ψ is uniformly continuous and |mBMS| < ∞, BMS + − − lim m (Ψ Ψ )=0. →0 Therefore by Proposition 3.4, we have that (3.9) holds for all non-negative bounded measurable f with compact support on E. Since the set of such f’s is dense in 1 PS ∈ 1 PS L (E,μE ) and both sides of (3.9) are linear and continuous in f L (E,μE ), ∈ 1 PS (3.9) holds for all f L (E,μE ). The following result is one of the basic tools developed in this article. ∈ 1 PS Theorem 3.5 (Transversal equidistribution). Let f L (E,μE ) be such that PS + − − → → ∈ § μE (f f ) 0 as 0.Letψ Cc(T ) for a transversal T of a box B ( 2.4). Then PS −δr −r μE (f) (3.26) lim e #(Γ¯r,t) · ψ(t) · f(G (t)) = · λT (ψ), r→∞ |mBMS| t∈T ∩Gr (E) where + − f (x)= sup f(y) and f (x) = inf f(y), {y∈E:d(y,x)< } {y∈E:d(y,x)< }

the transverse measure λT is defined by (3.6) and Γ¯r,t is defined by (2.27). Proof. Since both sides of (3.26) are linear in f and in ψ, without loss of generality we may assume that f ≥ 0andψ ≥ 0. By Lemma 2.17, supp(ψ)canbecovered by finitely many admissible boxes. By a partition of unity argument, in view of Remark 2.5, we may assume without loss of generality that T is a transversal of an admissible box B.

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Let 0 > 0 be such that p is injective on B 0+ and that ψ vanishes outside T 0−. We extend ψ to a continuous function on T 0+ by putting ψ =0onT 0+ T .Since B is admissible, due to Lemma 2.15, if we deﬁne PS ∈ × Ψ(tp)=ψ(t)/μH+ (tP ), for all (t, p) T 0+ P, t ± ∈ then Ψ is a bounded continuous function on B 0+ vanishing outside B 0−.IfΨ C(B +) are deﬁned as in (3.12) for 0 <≤ 0,then + − − (3.27) lim Ψ Ψ ∞ =0. →0 By Proposition 3.4, μPS(f +)mBMS(Ψ+) + Gr + PS ≤ E lim supr→∞ E Ψ ( (v))f (v) dμE (v) |mBMS| , (3.28) − − − − μPS(f )mBMS(Ψ ) Gr PS ≥ E lim infr→∞ E Ψ ( (v))f (v) dμE (v) |mBMS| .

BMS ∞ BMS + − − → → Since m (B 0+) < , by (3.27), we have that m (Ψ Ψ ) 0as 0. PS | ± − | → → By our assumption, μE ( f f ) 0as 0. Therefore by Proposition 2.18 and (3.28),

PS BMS −δr −r μE (f)m (Ψ) lim e #(Γ¯r,t) · ψ(t) · f(G (t)) = . r→∞ |mBMS| t∈T ∩Gr (E) Also, BMS PS m (Ψ) = dμT (t) Ψ(tp)dμH+ = λT (ψ). T tP t

Now we state and prove the main equidistribution result of this article which is more general than Theorem 1.8. ∈ 1 PS PS + − − → → Theorem 3.6. Let f L (E,μE ) such that μE (f f ) 0 as 0.Let 1 Ψ ∈ Cc(T (X)).Then μPS(f) lim e(n−1−δ)r Ψ(Gr(u))f(u) dμLeb(u)= E mBR(Ψ). →∞ E | BMS| r u∈E m

In particular, the result applies to f = χF for a Borel measurable F ⊂ E such PS ∞ PS that μE (F 1 ) < for some 1 > 0 and μE (∂F)=0. { PS } Proof. By Lemma 2.17, the boxes admissible with respect to μH+ form a basis of open sets in T1(X). By a partition of unity argument, without loss of generality we may assume that supp(Ψ) ⊂ B for an admissible box B = TP.Let0 > 0be ≤ ± such that Ψ = 0 outside B 0−.For0< 0,letΨ be deﬁned as in (2.18). Then + − − (3.29) lim Ψ Ψ ∞ =0. →0 ∈ ± ± Leb ± ∈ For t T 0 and >0, deﬁne ψ (t)= tP Ψ dμH+ . By Lemma 2.15, ψ t Cc(T ) for any 0 <<0/2.

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For the conformal density, {μx} = {mx},wehaveδμ = n−1, and by multiplying −δr all the terms in the conclusion of Corollary 2.14 by e ,forr>r (see (2.17)), we get −(n−1) −δr ¯ · − · − G−r (e )e #(Γr,t) ψ (t) f ( (t)) ∈ ∩Gr t T (E) ≤ (n−1−δ)r Gr Leb e Ψ( (u))f(u) dμE (u) E ≤ (n−1) −δr ¯ · + · + G−r (e )e #(Γr,t) ψ (t) f ( (t)). t∈T ∩Gr (E) Leb ∈ BR Deﬁne ψ(t):= tP Ψ(tp) dμH+ for all t T .ThenλT (ψ)=m (Ψ) and t ± BR ± BMS ∞ λT (ψ )=m (Ψ ). Since m (B 0+) < , by (3.29), + − − BR + − − → → λT (ψ ) λT (ψ )=m (Ψ Ψ ) 0, as 0. PS + − − → Also, since μE (f f ) 0, by Theorem 3.5 μPS(f)λ (ψ) lim e(n−1−δ)r Ψ(Gr(u))f(u) dμLeb(u)= E T . →∞ E | BMS| r E m BR Since λT (ψ)=m (Ψ), we prove the claim. In the particular case of f = χF ,wehave + − inf f = χF and sup f = χint(F ), and >0 >0 PS + PS PS + − PS ∞ → − if μE (f 1 )=μE (F 1 ) < , then lim 0 μE (f f )=μE (∂F). The idea of the above proof was inﬂuenced by the work of Schapira [34]. Our proof also yields the following variation of Theorem 3.6. Theorem 3.7. Let F˜ ⊂ E˜ be a Borel subset such that μPS(F˜ ) < ∞ for some >0 E˜ and μPS(∂F˜)=0. Then for any ψ ∈ C (T1(Γ\Hn)), E˜ c μPS(F˜) lim e(n−1−δ)t · ψ(Gt(v)) dμLeb(v)= E˜ · mBR(ψ). → ∞ E˜ | BMS| t + F˜ m

3.6. Integrability of the base eigenfunction φ0.

Proof of Theorem 1.17. We want to prove equivalence of the following: 1 n (1) φ0 ∈ L (Γ\H ,dVolRiem); (2) |mBR| < ∞; (3) Γ is a lattice in G. The pushforward of mBR from Γ\ T1(H1)toΓ\Hn is the measure corresponding to φ0 d VolRiem (see [17, Lemma 6.7]). Therefore (1) and (2) are equivalent. To prove that (2) implies (3), suppose that |mBR| < ∞. Since the left G-action on T1(Hn) is transitive, we may identify T1(Γ\Hn)withΓ\G/M for a compact subgroup M. We lift the measure mBR to a measure m on Γ\G as follows: for ∈ \ BR ¯ ¯ any f Cc(Γ G), we deﬁne m(f)=m (f), where f(ΓgM)= x∈M f(Γgx) dx, where dx is the probability Haar measure on M.DenotebyU the horospherical subgroup of G whose orbits in G project to the unstable horospheres in T1(Hn).

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Then M normalizes U and any unimodular proper closed subgroup of G containing U is contained in the subgroup MU.Asm is invariant under GH+ for any unstable horosphere H+, it follows that m is a U-invariant finite measure on Γ\G.By Ratner’s theorem [30], any ergodic component, say, λ,ofm is a homogeneous measure in the sense that λ is an H-invariant finite measure supported on a closed orbit x0H for some x0 ∈ Γ\G and a unimodular closed subgroup H of G containing U.IfH = G,thenH ⊂ MU and Γ ∩ H is cocompact in H. It follows by a theorem of Bieberbach ([4, Theorem 2.25]) that Γ ∩ U is cocompact in U. Hence H = U. Thus we can write m = m1 + m2,wherem1 is G-invariant and m2 is supported on a union of compact U-orbits. BR If m1 =0,thenm = m2, and hence the projection of the support of m in T1(Hn) is a union of compact unstable horospheres. It follows that the Patterson- Sullivan density is concentrated on the set of parabolic fixed points of Γ, which is a contradiction. If m1 =0,then m1 is a finite G-invariant measure on Γ\G; that is, Γ is a lattice in G. Hence (2) implies (3). BR If Γ is a lattice, then {νx} = {mx} up to a constant multiple. Hence m is the projection of a finite G-invariant measure of Γ\G to T1(Γ\Hn). Hence (3) implies (2).

4. Geometric finiteness of closed totally geodesic immersions 4.1. Parabolic fixed points and minimal subspaces. Let Γ be a torsion-free discrete subgroup of G. Definition 4.1. An element g ∈ G is called parabolic if Fix(g):={ξ ∈ ∂Hn : gξ = ξ} is a singleton set. An element ξ ∈ ∂Hn is called a parabolic fixed point of Γif there exists a parabolic element γ ∈ Γ such that Fix(γ)={ξ}.Notethatifξ is a parabolic fixed point for Γ, then ξ ∈ Λ(Γ). Let Λp(Γ) denote the set of parabolic fixed points of Γ.

n Let ξ ∈ Λp(Γ). In order to analyze the action of Γξ on ∂H {ξ},itisconvenient Rn { ∈ Rn−1 } Hn to use the upper half space model + = (x, y):x ,y >0 for ,whereξ ∞ Hn { } Rn { ∈ Rn−1} corresponds to and ∂ ξ corresponds to ∂ + = (x, 0) : x .The n ∼ subgroup Γ∞ acts properly discontinuously via affine isometries on ∂H {∞} = Rn−1. AtthisstagewewilltreatRn−1 only as an affine space, and we will choose its n−1 origin 0 later. Moreover, the action of Γ∞ preserves every horosphere R ×{y}, where y>0, based at ∞. By a theorem of Bieberbach ([4, 2.2.5]), Γ∞ contains a normal abelian subgroup of finite index, say Γ∞. By [4, 2.1.5], any (non-empty) Γ∞-invariant affine subspace n−1 of R contains a (non-empty) minimal Γ∞-invariant affine subspace; we call such an affine subspace a Γ∞-minimal subspace. By [4, 2.2.6], Γ∞ acts cocompactly via translations on any Γ∞-minimal subspace. Moreover, any two Γ∞-minimal subspaces are parallel, and if v1 and v2 belong to any two Γ∞-minimal subspaces, then γv1 − γv2 = v1 − v2 for all γ ∈ Γ∞.Letrank(Γ∞) denote the rank of the (torsion-free) Z-module Γ∞; it is independent of the choice of Γ∞, and it equals the dimension of a Γ∞-minimal subspace.

Deﬁnition 4.2. A parabolic ﬁxed point ξ ∈ Λp(Γ) is said to be bounded if Γξ\(Λ(Γ) {ξ})iscompact.DenotebyΛbp(Γ) the set of all bounded parabolic

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fixed points for Γ. Therefore ∞∈Λbp(Γ) if and only if ∞∈Λp(Γ) and n−1 (4.1) Λ(Γ) {∞}⊂{x ∈ R : dEuc(x, L) ≤ r0}, for some r0 > 0, where L is a Γ∞-minimal subspace. ˜ 4.2. On geometric finiteness of ΓS˜. For the rest of this section, let S be a proper connected totally geodesic subspace of Hn such that the natural projection map \ ˜ → \Hn \ → \ ΓS˜ S X =Γ is proper or, equivalently, the map ΓS˜ GS˜ Γ G is proper ˜ or, equivalently, ΓGS˜ is closed in G.SinceS is totally geodesic, the geometric boundary ∂S˜ is the intersection of ∂Hn with the closure of S˜ in Hn. Proposition 4.3. Let ∞∈Λp(Γ) ∩ ∂S˜.LetL be a Γ∞-minimal subspace of ∼ − ∂Hn {∞} = Rn 1 and choose the origin 0 ∈ L. Then the intersection of L with ˜ {∞} ∩ the (parallel) translate of the affine subspace ∂S through 0 is a (Γ∞ GS˜)- minimal subspace. ∩ ˜ {∞} Proof. Let Γ =Γ∞,Δ=Γ GS˜, and the affine subspace F = ∂S .Since ΔF = F ,letv belong to a Δ-minimal subspace of F .Sincev and 0 belong to two Δ- minimal subspaces, γv−γ0=v−0=v.Sinceγv ∈ F ,wehaveγ0=γv−v ∈ F −v. Since 0 ∈ F − v,wehaveγ0 ∈ γ(F − v) ∩ (F − v). Now γ(F − v)andγF = F are parallel. Therefore F − v and γ(F − v) are parallel, and since they intersect, γ(F − v)=F − v.ThusΔ(F − v)=F − v. Therefore, Δ-action preserves ∩ − ∩ L0 := L (F v). We want to prove that Γ GS˜ acts cocompactly on L0. Since ∞∈Λp(Γ), by [4, Lemma 3.2.1] Γ∞ consists of parabolic elements of G∞.Thatis,Γ∞ ⊂ MN,whereN is the maximal unipotent subgroup of G which acts transitively on Rn−1 = ∂Hn {∞} via translations and M is a compact subgroup of G normalizing N that acts on Rn−1 by Euclidean isometries fixing 0. Let U = {g ∈ N : gL = L}.ThenU acts transitively on L by translations. Since 0 ∈ L and Γ acts cocompactly on L via translations, the connected component of the Zariski closure of Γ in G is a connected abelian subgroup of the form MLU, where ML ⊂ M and ML acts trivially on L. Since Γ \L is a compact Euclidean torus, the closure of the image of L0 in Γ \L equals the image of an affine subspace, say L1,ofL.ThusΓ L0 =ΓL1. For i =0, 1, let Ui = {u ∈ U : uLi = Li}.ThenUi acts transitively on Li,and Γ MLU0 =ΓMLU1. Therefore the identity component of Γ U0 is of the form M1U1, where M1 ⊂ ML and (Γ ∩ M1U1)\M1U1 is compact. In particular, Γ ∩ M1U1 acts cocompactly on L1. ⊂ By our assumption ΓGS˜ is a closed subset of G. Therefore Γ U0 ΓGS˜.SinceGS˜ ⊂ is the identity component in ΓGS˜,wehaveM1U1 GS˜. It follows that U1 preserves L0.SinceU1 acts transitively on L1, L1 ⊂ L0; hence L1 = L0.Inparticular, ∩ ∩ Γ M1U1 acts cocompactly on L0. Therefore Δ = Γ GS˜ acts cocompactly on L0. Proposition 4.4. Let ∞∈Λ (Γ) ∩ ∂S˜ and Γ := Γ ∩ G .Then bp S˜ S˜ ∞∈ ∩ Λbp(ΓS˜) if Γ∞ ΓS˜ is infinite, ∞ ∈ ∩ / Λ(ΓS˜) if Γ∞ ΓS˜ is finite, hence trivial.

Proof. Let the notation be as in Proposition 4.3. Since ∞∈Λbp(Γ), by (4.1), Λ(Γ) {∞} is contained in a bounded neighborhood of L, and hence in a bounded neighborhood of L + v. Therefore (Λ(Γ) {∞}) ∩ ∂S˜ is contained in a bounded

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neighborhood of L + v intersected with F = ∂S˜ {∞}, and hence in a bounded ∩ − ∩ neighborhood of L0 = L (F v) as well. By Proposition 4.3, L0 is a (ΓS˜ Γ )- ∩ ∞∈ minimal subspace. Now if Γ∞ ΓS˜ is inﬁnite, or equivalently Λp(ΓS˜), then ∞∈ Λbp(ΓS˜). ∩ {∞} Suppose that Γ∞ ΓS˜ is ﬁnite. Then L0 is a singleton set. Therefore Λ(ΓS˜) ˜{∞} ∞∈ ˜ is contained in a bounded subset of ∂S .Then ∂S is isolated from Λ(ΓS˜). Since the limit set of a non-elementary hyperbolic group is perfect, it follows that

ΓS˜ is elementary, and hence ΓS˜ is either parabolic or loxodromic. Now suppose ∞∈ {∞} that Λ(ΓS˜). In the parabolic case Λ(ΓS˜)= =Λp(ΓS˜), contradicting the ∩ { } ∞∈ ⊂ assumption that Γ∞ ΓS˜ = e . In the loxodromic case, Λr(ΓS˜) Λr(Γ), contradicting the assumption that ∞∈Λp(Γ). Lemma 4.5. We have ∩ ˜ Λr(Γ) ∂S =Λr(ΓS˜).

Proof. Let ξ ∈ Λr(Γ) ∩ ∂S˜.AsS˜ is totally geodesic, there exists a geodesic ray, say, β,lyinginS˜ pointing toward ξ.Sinceξ is a radial limit point, Γβ accumulates on a compact subset of Hn. By the assumption that the natural projection map \ ˜ → ˜ ΓS˜ S X is proper, ΓS˜β accumulates on a compact subset of S. This implies ∈ ξ Λr(ΓS˜). The other direction for the inclusion is clear. In [4], Bowditch proved the equivalence of several deﬁnitions of geometrically ﬁnite hyperbolic groups. In particular, we have:

Theorem 4.6 ([2], [4], [21]). Γ is geometrically ﬁnite if and only if Λ(Γ) = Λr(Γ)∪ Λbp(Γ).

Hence, for geometrically ﬁnite Γ, we have Λp(Γ) = Λbp(Γ).

Theorem 4.7. If Γ is geometrically ﬁnite, then ΓS˜ is geometrically ﬁnite.

Proof. Since Λ(Γ) = Λr(Γ)∪Λbp(Γ), it follows from Proposition 4.4 and Lemma 4.5 ∪ that Λ(ΓS˜)=Λbp(ΓS˜) Λr(ΓS˜), proving the claim by Theorem 4.6. PS 4.3. Compactness of supp(μE ) for horospherical E. Theorem 4.8 (Dal’bo [7]). Let Γ be geometrically finite. For a horosphere H in T1(Hn) based at ξ ∈ ∂Hn, E := p(H) is closed in T1(X) if and only if either ξ/∈ Λ(Γ) or ξ ∈ Λp(Γ). Theorem 4.9. Let Γ be geometrically finite. If E := p(H) is a closed horosphere 1 PS in T (X),thensupp(μE ) is compact. Proof. Let ξ ∈ ∂Hn bethebasepointforH. The restriction of the visual map Vis : v → v+ induces a homeomorphism ψ : H→∂Hn {ξ}.AsE is closed, by Theorem 4.8 either ξ/∈ Λ(Γ) or ξ is a bounded parabolic fixed point. If ξ/∈ Λ(Γ), Hn { } PS −1 then Λ(Γ) is a compact subset of ∂ ξ . Since supp(μE )=p(ψ (Λ(Γ))), it PS follows that supp(μE )iscompact. Suppose now that ξ is a bounded parabolic fixed point. By Definition 4.2, Γξ\(Λ(Γ) {ξ})iscompact.SinceΓξ is discrete, it preserves the horosphere H based at ξ,andΓξ =ΓH. Therefore ψ induces a homeomorphism between ΓH\H n −1 and ΓH\(∂H {ξ}). It follows that ΓH\ψ (Λ(Γ) {ξ}) is compact and is equal PS to supp(μE ).

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5. On the cuspidal neighborhoods of Λbp(Γ) ∩ ∂S˜ 5.1. Throughout this section, let Γ be a torsion-free discrete subgroup of G and S˜ a connected complete totally geodesic subspace of Hn such that the natural projection \ ˜ → \Hn ΓS˜ S Γ is a proper map. ∈ ˜ The Dirichlet domain for ΓS˜ attached to some a S is deﬁned by D { ∈ ˜ ≤ ∈ } (5.1) (a, ΓS˜):= s S : d(s, a) d(s, γa) for all γ ΓS˜ . ∩ D ∅ Proposition 5.1. Λr(Γ) ∂ (a, ΓS˜)= . ∈ ∩ D Proof. Let ξ Λr(Γ) ∂ (a, ΓS˜). As D D ∪ D ∩ Hn (a, ΓS˜)= (a, ΓS˜) (∂ (a, ΓS˜) ∂ ) Hn { }⊂D is convex in , there exists a geodesic ξt (a, ΓS˜) such that ξ0 = a and ∈ →∞ ∈ ξ∞ = ξ.Asξ Λr(ΓS˜) by Lemma 4.5, there exist sequences ti and γi ΓS˜ →∞ such that d(γiξti ,a) is uniformly bounded for all i.Sinced(ξti ,a) , it follows −1 ∈D that for all large i, d(ξti ,γi a)

Let E˜ ⊂ T1(Hn) denote the set of all normal vectors to S˜.GivenU ⊂ ∂Hn,we deﬁne E { ∈ ˜ ∈D + ∈ ∩ } (5.2) U = v E : π(v) (a, ΓS˜),v U Λ(Γ) .

n Remark 5.2. If ξ ∈ Λr(Γ) ∩ ∂S˜, then there exists a neighborhood U of ξ in ∂H such that EU = ∅. To see this, note that if there exists a sequence {vi}⊂E˜ such + → → ∈D that vi ξ,thenπ(vi) ξ, and hence by Proposition 5.1 π(vi) (a, ΓS˜) for all large i. In view of Theorem 4.6 and Remark 5.2, the main goal of this section is to describe the structure of EU for a neighborhood U of a point in Λbp(Γ) ∩ ∂S˜ and to compute the measure μPS(E ). E˜ U n n−1 In this section, we will use the upper half space model H = R × R>0,and ﬁrst we assume that

∞∈∂S˜ ∩ Λp(Γ). Here Rn−1 is to be treated as an affine space until we make a choice of the origin. Hence S˜ is a vertical plane over the affine subspace ∂S˜ {∞} of Rn−1. For any n−1 n−1 affine subspace F of R ,letPF : R → F denote the orthogonal projection. Let

n−1 n−1 n−1 (5.3) b : R × R>0 → R and h : R × R>0 → R>0 denote the natural projections. Let Γ =Γ∞ be a normal abelian subgroup of Γ∞ with finite index, as in §4.1, and fix a Γ-minimal subspace L of Rn−1. Noting that b(a) ∈ ∂S˜ {∞},wechoose n−1 0:=PL(b(a)), the origin of R . This choice of 0 makes L a linear subspace. Set { − ∈ ˜ {∞}} Rn−1 ∩ W := v b(a):v ∂S , a linear subspace of ,andΔ:=ΓS˜ Γ .By Proposition 4.3, L0 := L ∩ W is a Δ-minimal (linear) subspace. Let V be the largest affine subspace of Rn−1 such that Δ acts by translations on V .Then0∈ L ⊂ V and V is the union of all (parallel) Δ-minimal subspaces of

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n−1 n−1 R . There exist group homomorphisms τ :Δ→ L0 ⊂ R and θ :Δ→ O(n−1) such that for any γ ∈ Δ, (5.4) γ(x)=θ(γ)(x)+τ(γ), for all x ∈ Rn−1. We note that V = {x ∈ Rn−1 : θ(Δ)x = x},andV ⊥ is the sum of all non-trivial (two-dimensional) θ(Δ)-irreducible subspaces of Rn−1. Lemma 5.3. (1) W =(W ∩ V )+(W ∩ V ⊥); (2) W ⊥ =(W ⊥ ∩ V )+(W ⊥ ∩ V ⊥). Proof. Put F = ∂S˜ {∞}.ThenΔF = F , and there exists a Δ-minimal aﬃne ⊂ ∈ ⊂ ∩ subspace LS˜ F .Choose0 LS˜ F V .SinceW is a parallel translate of F through 0, we have W = F − 0. As in the proof of Proposition 4.3, Δ(W )=W . Since 0 ∈ W ,wehaveθ(Δ)(W )=W , and hence θ(Δ)(W ⊥)=W ⊥.ThusW ∩ V is the set of ﬁxed points of θ(Δ) in W , and its orthocomplement in W is the sum of all non-trivial θ(Δ)-irreducible subspaces of W anditisthesameasW ∩ V ⊥. Therefore (1) follows, and (2) is proved similarly. For any v ∈ E˜, π(v) ∈ S˜. By abuse of notation, we write b(v):=b(π(v)) ∈ ⊥ ∂S˜ {∞} and h(v):=h(π(v)) ∈ R>0.Wedenotebyσ(v) ∈ W the unique element in W ⊥ of norm one satisfying (5.5) v+ := Viz(v)=b(v)+h(v)σ(v). Bounded parabolic assumption. For the rest of this section we will further assume n−1 that ∞∈∂(S˜)∩Λbp(Γ). Hence there exists R0 > 0 such that for all x ∈ Λ(Γ)∩R ,

(5.6) PL⊥ (x)≤R0, where · denotes the Euclidean norm. Lemma 5.4. For any v ∈ E˜ with v+ ∈ Λ(Γ),

PV ⊥ (b(v))≤R0. Proof. Let 0 ∈ V be as in the proof of Lemma 5.3. Since b(v)−0 ∈ W and 0 ∈ V , we have PV ⊥ (0 ) = 0, and by Lemma 5.3, ⊥ PV ⊥ (b(v)) = PV ⊥ (b(v) − 0 ) ∈ W and PV ⊥ (σ(v)) ∈ W . + ⊥ ⊥ Therefore by (5.5), PV ⊥ (b(v))≤PV ⊥ (v ).SinceL ⊂ V ,wehaveV ⊂ L , + + and hence by (5.6), PV ⊥ (v )≤PL⊥ (v )≤R0.

Proposition 5.5. There exists R1 > 0 such that for all v ∈E∂Hn , + ≤ PL0 (v ) R1. ∈E ∈ ⊂ Proof. Let v ∂Hn . Then for all γ Δ ΓS˜,

(5.7) dhyp(π(v),a) ≤ dhyp(γπ(v),a) ⇒ deucl(b(v), b(a)) ≤ deucl(γ b(v), b(a)). − ∈ ⊂ ∩ Now b(v) b(a) W , L0 W V ,andPL0 (b(a)) = 0. As ⊥ ∩ ∩ ∩ ⊥ W =(V W )+L0 +(W V L0 ), which is a sum of θ(Δ)-invariant orthogonal subspaces of W ,weget

γ b(v) − b(a)=[θ(γ)PV ⊥ (b(v)) − PV ⊥ (b(a))]

+[PL (b(v)) + τ(γ)] + P ∩ ∩ ⊥ (b(v) − b(a)). 0 V W L0

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Comparing this with (5.7), for any γ ∈ Δweget 2 2 2 ≤ ⊥ − ⊥ (5.8) PL0 (b(v)) θ(γ)PV (b(v)) PV (b(a)) + PL0 (b(v)) + τ(γ) .

Since τ(Δ) is a lattice in L0, the radius of the smallest ball containing a fundamental domain of τ(Δ) in L0 is ﬁnite, which we denote by R2. Then by (5.4) and (5.8), we conclude that + 2 2 ≤ 2 2 PL0 (v ) = PL0 (b(v)) (R0 + b(a) ) + R2. 2 2 1/2 By setting R1 =((R0 + b(a) ) + R2) , we ﬁnish the proof.

5.2. Corank at ∞ and the structure of EU . Set − ∩ r∞ := rank(Γ∞) rank(Γ∞ ΓS˜). More precisely, r∞ =rank(Γ) − rank(Δ) = dim(L) − dim(L0). n Proposition 5.6. If r∞ =0, then there exists a neighborhood U of ∞ in ∂H such that EU = ∅,whereEU is deﬁned in (5.2). n−1 Proof. As r∞ =0,wehaveL = L0. Therefore, for all x ∈ Λ(Γ) ∩ R ,

P ⊥ (x)≤R0. L0

Hence for any v ∈E∂Hn , by Proposition 5.5, + 2 + 2 + 2 2 2 v = PL (v ) + P ⊥ (v ) ≤ R + R . 0 L0 1 0 { ∈ Rn−1 2 2 2}∪{∞} E ∅ Let U = x : x >R0 + R1 .Then U = . In the rest of this section, we now consider the case when

r := r∞ ≥ 1. r Notation 5.7. For any s =(s1,...,sr) ∈ R and an ordered r-tuple (w1,...,wr)of n−1 n−1 r r vectors in R ,wesets · w := s1w1 + ···+ srwr ∈ R , R w := {s · w : s ∈ R } r and |s| =max(|s1|,...,|sr|). For k ∈ Z and an ordered r-tuple γ =(γ1,...,γr) k k1 ··· kr ∈ of elements of G,wewriteγ = γ1 γr G. Fix an ordered r-tuple γ =(γ1,...,γr)ofelementsofΓ =Γ∞ such that the subgroup generated by γ ∪Δ is of ﬁnite index in Γ .Foreachγi,thereexistswi ∈ L n−1 and σi ∈ O(n − 1) such that for all x ∈ R ,

γi(x)=σi(x)+wi. ∈ Z k Moreover, σi and the translation by wi commutes, and hence for any k , γi (x)= k σi (x)+kwi. Setting w =(w1,...,wr)andσ =(σ1,...,σr), we have that for any x = y + z ∈ n−1 ⊥ r R with y ∈ L and z ∈ L and k =(k1,...,kr) ∈ Z , (5.9) γk(x)=σk(y)+z + k · w.

Let R0 and R1 be as in (5.6) and in Proposition 5.5, respectively. Set ⊥ B0 := {x ∈ L : x≤R0} and B1 := {x ∈ L0 : x≤R1}. ∩ ⊥ Zr · Rr Let M1 := L L0 .Then PM1 (w) is a lattice in M1 = PM1 (w), which admits a relatively compact fundamental domain, say F1.LetF2 be a relatively compact fundamental domain for the lattice τ(Δ) in L0. We deﬁne the following relatively compact subset of Rn−1: F ⊂ ⊥ ∩ ⊥ (5.10) := B0 +(B1 + F2)+F1 L + L0 +(L L0 ).

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By (5.9), γkF = F + k · w. For related variable quantities x ≥ 0andy ≥ 0, the symbol x y means that there exists a constant C>0 such that for all related x and y, x ≥ Cy,andthe symbol x y means that x y and y x.

Proposition 5.8. There exists c0 ≥ 1 such that for all suﬃciently large N ≥ 1, (5.11) Viz(E ) ⊂ Δγk(F), Uc0N |k|≥N { ∈ Rn−1 ≥ } where Uc0N = x : x c0N . Rn−1 ⊥ ⊥ ∩ ∈ Rn−1 Proof. Since = L + L0 + M1 for M1 = L0 L, we have for any v , + + + + ⊥ (5.12) v = PL (v )+PL0 (v )+PM1 (v ).

Let v ∈E∂Hn . By (5.6) and Proposition 5.5, + + ⊥ ∈ ∈ (5.13) PL (v ) B0 and PL0 (v ) B1. + + ∈ Zr + ∈ In order to control PM1 (v ), let k = k(v ) be such that PM1 (v ) · ∈ k PM1 (w)+F1,wherek is uniquely determined. Let λk Δ be such that · ∈ PL0 (k w) τ(λk)+F2. · − · · Since k PM1 (w) k w = PL0 (k w), + ∈ · − · · ∈ · PM1 (v ) (k PM1 (w) k w)+(F1 + k w) τ(λk)+F2 +(F1 + k w). Therefore by (5.12), for k = k(v+), we have

+ k (5.14) v ∈F+ k · w + τ(λk)=λkγ (F). Rr → ≥ Since PM1 : w M1 is a linear isomorphism, there exists N1 1 such that r for all k ∈ Z with |k| >N1, · | | (5.15) PM1 (k w) k . + − +≤ + − · ∈ By (5.12) and (5.13), PM1 (v ) v R0 +R1 and PM1 (v ) PM1 (k w) F1 for k = k(v+). It follows that there exists a constant B>0 such that for all v ∈E∂Hn , · − ≤ +≤ · PM1 (k w) B v PM1 (k w) + B, + where k = k(v ). Hence by (5.15), there exists N2 ≥ 1 such that for all v ∈E∂Hn + with |k(v )| >N2, v+|k(v+)|. In view of (5.14), this ﬁnishes the proof.

r Lemma 5.9. There exists N0 ≥ 1 such that for all k ∈ Z with |k| >N0,the following hold: (1) For ξ ∈ γk(F), ξ|k|. (2) For v ∈ E˜ with v+ ∈ γk(F), h(v) |k|.

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k Proof. If ξ ∈ γ F,thenξ − k · w≤diameucl(F). Hence ξk · w|k|, proving (1). For v ∈ E˜ such that v+ ∈ γk(F), by (5.5), + h(v) PW ⊥ (v )PW ⊥ (k · w). r r ⊥ Since W ∩ L = L0 and L = L0 ⊕ R w,themapPW ⊥ : R w → W is injective. Therefore

PW ⊥ (k · w)k · w|k|, from which (2) follows.

n−1 Let o =(0, 1) ∈ R × R>0.ForT ≥ 1, put

(5.16) BT = {v ∈ E˜ : β∞(o, π(v)) ≥ log T };

that is, BT is the intersection with E˜ of a horoball based at ∞. We note that for v ∈ E˜, β∞(o, π(v)) = log h(v). Hence in the vertical plane model of S˜, BT consists of vectors v ∈ E˜ whose base points have the Euclidean height at least T .

Proposition 5.10. Let F0 := B0 + F2 + F1.Thenνo(F0) > 0,andforallsuﬃ- ciently large T ,thereexistsN T such that k (5.17) Viz(BT ) ⊃ γ (F0). |k|≥N

r Proof. Since ΔF = L and γZ F = M , 2 0 1 1 k Δ( γ (F0)) = B0 + L0 + M1 = B0 + L ⊃ Λ(Γ) {∞}. k∈Zr

Therefore, if νo(F0) = 0, then by the conformality, it follows that νo(Λ(Γ){∞})= 0. Since Γ does not ﬁx ∞, by the Γ-invariance of {νx} we get νo(Λ(Γ)) = 0, which is a contradiction, proving the ﬁrst claim. + k If v ∈ E˜ and v ∈ γ (F0), then by Lemma 5.9, h(v) |k|.Ifh(v) >T,then v ∈BT . Therefore (5.17) holds for suitable N T . 5.3. Estimation of μPS(E ). Let V−1 : Rn−1 ∂S˜ → E˜ be the inverse of the E˜ U restriction of the visual map Viz : E˜ → ∂Hn ∂S˜ = Rn−1 ∂S˜. Lemma 5.11. There exists N ≥ 1 such that for all k ∈ Zr with |k| >N , 1 1 −1 δβξ(o,π(V (ξ))) −δ e dνo(ξ) |k| . ξ∈γkF

Proof. We have PW ⊥ (k · w)|k|. Hence for suﬃciently large |k|,wehavethat γkF∩∂S˜ = ∅. Note that the Euclidean diameter of the horosphere based at ξ n−1 2 and passing through o =(0, 1) ∈ R × R>0 is 1 + ξ , and the diameter of the horosphere based at ξ and passing through π(V−1(ξ)) is h(π(V−1(ξ))). Therefore the signed hyperbolic distance of the segment cut by these two horospheres on the vertical geodesic ending in ξ is −1 2 −1 βξ(o, π(V (ξ))) = log(1 + ξ ) − log(h(π(V (ξ)))). Hence by Lemma 5.9, 2 −1 1+ξ δ (5.18) eδβξ (o,π(V (ξ))) = |k|δ. h(π(V−1(ξ)))

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By conformality and Γ-invariance of Patterson-Sullivan density {ν }, x − k −k dνγ k·o ν (γ F)=(γ ν )(F)=ν −k· (F)= (ξ) dν (ξ) o o γ o dν o ξ∈F o −k −δβξ(γ o,o) (5.19) = e dνo(ξ). ξ∈F We note that the horosphere based at ξ passing through γ−ko =(−k · w, 1) ∈ n−1 2 R × R>0 has diameter 1 + ξ + k · w . Therefore −k 2 2 βξ(γ o, o)=log(1+ξ + k · w ) − log(1 + ξ ), and hence, since k · w|k| for all large |k|,wehave,foranyξ ∈F, 2 − − k 1+k · w − ξ δ − e δβξ(γ o,o) = |k| 2δ. 1+ξ2

Since νo(F) > 0 by Proposition 5.10, we deduce from (5.19) that k −2δ −2δ νo(γ F) |k| νo(F) |k| . Together with (5.18), this proves the claim. ˜ → \ ˜ Let p : E ΓE˜ E be the natural quotient map. We note that ΓE˜ =ΓS˜.From §2.6, we recall that the measure μPS, which is Γ -invariant, naturally induces a E˜ E˜ \ ˜ \ ˜ ˜ measure on ΓE˜ E. The pushforward of this measure from ΓE˜ E to E = p(E)is PS μE .

Recall the definition of c0 > 0andUc0N from Proposition 5.8. Proposition 5.12. (1) For all sufficiently large N ≥ 1, we have μPS(p(E )) |k|−δ. E Uc0N k∈Zr {0} (2) For all sufficiently large T ≥ 1, we have PS B | |−δ μE (p( T )) k . k∈Zr {0} Proof. By Proposition 5.8 and Lemma 5.11, for all large N ≥ 1, μPS(p(E )) ≤ μPS(V−1(γk(F))) E Uc0N E˜ |k|≥N −1 δβξ(o,π(V (ξ))) = e dνo(ξ) | |≥ ξ∈γkF kN |k|−δ, k≥N proving (1). Consider the natural quotient map ∩ \ ˜ → \ ˜ (5.20) p∞ :(ΓE˜ Γ∞) E ΓE˜ E. ∞∈ ∩ \B Since Λbp(Γ), there exists T0 > 0 such that p∞ restricted to (ΓE˜ Γ∞) T is proper and injective for all T ≥ T0. Now since F2 is a fundamental domain for Δ action on LS˜ and F1 is a funda- k r mental domain for the action of {γ : k ∈ Z } on M1, the quotient map E˜ → Δ\E˜

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 544 HEE OH AND NIMISH A. SHAH −1 k F ∩ ∞ is injective on |k|≥N V (γ ( 0)). Since [ΓS˜ Γ∞ :Δ]< and p∞ is injective ∩ \B on (ΓS˜ Γ∞) T , for all sufficiently large T 1, PS B PS B μE (p( T )) = μ \ ˜ (p∞( T )); see (2.13) ΓE˜ E μPS(V−1(γk(F ))); by Proposition 5.10 |k|≥N E˜ 0 | |−δ k≥N k ; by Lemma (5.11). This proves (2). Parabolic corank and criterion for finiteness of PS 6. μE Let Γ be non-elementary torsion-free discrete subgroup of G.LetS˜, E˜ and E be § ˜ \ ˜ → \Hn as in 5. In particular, S is totally geodesic and the map ΓS˜ S Γ is proper. Definition 6.1 (Parabolic corank). Define − ∩ pb-corank(ΓS˜)= max (rank(Γξ) rank(Γξ ΓS˜)) . ξ∈Λp(Γ)∩∂(S˜) ∩ ˜ ∅ When Λp(Γ) ∂(S)= , we set pb-corank(ΓS˜)=0. ≤ ˜ Lemma 6.2 (Corank Lemma). pb-corank(ΓS˜) codim(S). n Proof. Suppose ∞∈Λp(Γ) ∩ ∂S˜.LetL be a Γ∞-minimal subspace of ∂H ∞ and let W be the intersection of a translate of ∂S˜ {∞} through a point in − ∩ − ≤ L. Then by Proposition 4.3, rank(Γ∞) rank(ΓS˜ Γ∞)=dim(L) dim(W ) (n − 1) − dim(∂S˜)=n − dim S˜. 6.1. Finiteness criterion for geometrically finite Γ. For the rest of this section we further assume that Γ is geometrically finite. ⇔ PS Theorem 6.3. pb-corank(ΓS˜)=0 supp(μE ) is compact. PS D Proof. Suppose that supp(μE ) is not compact. Fix a Dirichlet domain (a, ΓS˜) ˜ \ ˜ \ 1 Hn for the ΓS˜ action on S. Since the projection of ΓE˜ E into Γ T ( ) is proper, ∈ ˜ ∈D + ∈ there exists an unbounded sequence vm E with π(vm) (a, ΓS˜)andvm Λ(Γ). + → Since Λ(Γ) is compact, by passing to a subsequence, we assume that vm ξ for n some ξ ∈ Λ(Γ). Thus for any neighborhood U of ξ in ∂H ,wehavevm ∈EU for all large m. n n−1 Consider the upper half space model H = R × R>0 with ξ identified with ∞ § + → ∞ →∞ →∞ as in 5. As vm ξ = , by (5.5) we have b(vm) or h(vm) →∞ ∈ D (see (5.3) for notation), and hence π(vm) = ξ. Therefore ξ ∂( (a, ΓS˜)). By Proposition 5.1, ξ ∈ Λr(Γ). Since Γ is geometrically finite, by Theorem 4.6, ∈ ∩ D ξ Λbp(Γ) ∂ (a, ΓS˜). Now by Proposition 5.6, pb-corank(ΓS˜) =0. To prove the converse, suppose that there exists ξ ∈ Λbp(Γ) ∩ ∂S˜ such that − ∩ ≥ r =rank(Γξ) rank(Γξ ΓS˜) 1. Without loss of generality, we may assume ∞ ∩ \B ξ = .FixT0 > 1. The map p∞ as in (5.20) restricted to (ΓE˜ Γ∞) T0 is \ ˜ proper (see (5.16) for notation). Therefore for any compact subset Ω of ΓE˜ E,we have p∞(B ) ∩ Ω=∅ for all sufficiently large T>T. By Proposition 5.12(2), T 0 PS B | |−δ μE (p( T )) k > 0. k∈Zr {0} PS B Therefore supp(μE ) intersects p( T ) for all large T 1. Since the projection of \ ˜ \ 1 Hn PS ΓE˜ E into Γ T ( )isproper,supp(μE )isnon-compact.

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⇔| PS| ∞ Theorem 6.4. pb-corank(ΓS˜) <δ μE < . ≥ ∈ ∩ ˜ Proof. Suppose that pb-corank(ΓS˜) δ>0. Then there exists ξ Λbp(Γ) S − ∩ ≥ { } such that r := rank(Γξ) rank(Γξ ΓS˜) max δ, 1 . Without loss of generality, we may assume ξ = ∞. By the second part of the proof of Theorem 6.3, for all sufficiently large T 1, since r ≥ δ, | PS|≥ PS B | |−δ ∞ μE μE (p( T )) k = . k∈Zr {0} ∩ D Now suppose that pb-corank(ΓS˜) <δ. By the compactness of Λ(Γ) ∂( (a, ΓS˜)), D PS where (a, ΓS˜) is a fixed Dirichlet domain for ΓS˜, to prove finiteness of μE it suf- ∈ ∩ D fices to show that for every ξ Λ(Γ) ∂( (a, ΓS˜)), there exists a neighborhood U of Hn PS E ∞ E ξ in ∂ such that μE (p( U )) < with U defined as in (5.2). By Proposition 5.1 ∈ − ∩ and Theorem 4.7, ξ Λbp(Γ). Let r := rank(Γξ) rank(Γξ ΓS˜). If r =0,thenby Proposition 5.6, there exists a neighborhood U of ξ such that EU = ∅. Therefore we assume that δ>r≥ 1. By Proposition 5.12(1), there exists a neighborhood U of ξ such that PS E | |−δ ∞ μE (p( U )) k < . k∈Zr {0} Leb PS 6.2. Finiteness of μE and μE . Theorem 6.5. Let S˜ be any totally geodesic immersion in Hn. Suppose that ˜ ≥ | Leb| ∞ | PS| ∞ dim(S) (n +1)/2 and μE < .Then μE < . ˜ Proof. Since ΓS˜ is a lattice in GS˜,Λ(ΓS˜)=∂S. Hence by Theorem 2.21, the \ ˜ → \Hn natural map p :ΓS˜ S Γ is proper. Let k := dim(S˜) ≥(n +1)/2≥2. By a property of a lattice in rank one Lie ∩ − § group GS˜,rank(ΓS˜ Γξ)=k 1 (cf. [29, 13.8]). Therefore by Lemma 6.2, ≤ − ≤ − ≤ − r := pb-corank(ΓS˜) n k n (n +1)/2 (n 1)/2. ∈ ˜ ∩ ∩ ≥ − Let ξ ∂(S) Λbp(ΓS˜) be such that rank(ΓS˜ Γξ)=r.Thenrank(Γξ) (k 1)+r. By a result of Dalbo, Otal and Peign [8, Proposition 2],

δ>rank(Γξ)/2 ≥ ((k − 1) + r)/2 ≥ (k − 1+(n − k))/2=(n − 1)/2 ≥ r. | PS| Hence by Theorem 6.4, μE is ﬁnite. As an immediate corollary, we state: | Leb| ∞ | PS| ∞ Corollary 6.6. Let n =2, 3.Then μE < implies that μE < .

To deduce that skΓ(w0) > 0, when w0Γ is inﬁnite in Theorem 1.2 we need the following. Here Γ need not be geometrically ﬁnite. ∞ ⊂ ˜ | PS| Proposition 6.7. If [Γ : ΓS˜]= ,thenΛ(Γ) ∂∞(S),and μE > 0.

Proof. Suppose on the contrary that Λ(Γ) ⊂ ∂∞(S˜). Let L be a geodesic joining two distinct points, say ξ1,ξ2 ∈ Λ(Γ). Then L ⊂ S˜. For any γ ∈ Γ, we have γL as the geodesic joining γξ1 and γξ2, and hence γL ⊂ S˜.Nowﬁxx0 ∈ L.Then ⊂ ˜ \ ˜ → \Hn \ Γx0 S. Since ΓS˜ S Γ is a proper map, we get that ΓS˜ Γ is ﬁnite, a contradiction to our assumption.

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7. Orbital counting for discrete hyperbolic groups As before, let G =SO(n, 1)◦ for n ≥ 2 and Γ a torsion-free, non-elementary, discrete subgroup of G.

7.1. Computation with m˜ BR. Let K be a maximal compact subgroup of G.Let ∈ Hn ∼ Hn ∈ 1 Hn o be such that K = Go.ThenG/K = .LetX0 To( )andM = GX0 . ∼ 1 n Then G/M = T (H ), where g[M]=gX0.LetA = {ar : r ∈ R}⊂ZG(M) be a one-parameter subgroup of G consisting of diagonalizable elements such that Gr → + ∼ Hn (X0)=ar[M]. Via the map k kX0 ,wehaveK/M = ∂ . Let N

Lemma 7.1. For any g ∈ G, consider the measure λg on N given by − (n 1)β + (o,gz(o)) + gzX0 ∈ dλg(z)=e dmo(gzX0 ), where z N.

Then λg = λe.Inparticular,λe is a Haar measure on N which we shall denote by the integral dn = dλe(n) on N

Proof. Since {mx} is a G-invariant conformal density,

−1 (n−1)β + (o,g (o)) + + zX + dmo(gzX0 )=dmg−1(o)(zX0 )=e 0 dmo(zX0 ). −1 Since β + (o, gz(o)) = β + (g (o),z(o)), gzX0 zX0

(n−1)β + (o,z(o)) zX + (7.2) dλg(z)=e 0 dmo(zX0 )=dλe(z).

For any g ∈ N, dλe(gz)=dλg(z)=dλe(z). Therefore λe is N-invariant.

− ∩ − ∈ Notation 7.2. Note that GX = ANM and K GX = M.Forψ C(K)anda − ∼0 0 measure λ on ∂Hn = KX = K/M, we deﬁne 0 − (7.3) ψ(k) dλ(kX0 ):= ψ(km) dm dλ(kM). k∈K K/M m∈M

n BR We also fix a Patterson-Sullivan density {νx} on ∂H and considerm ˜ defined as in §3.1 with respect to {mx} and {νx}. Proposition 7.3. For any φ ∈ C (T1(Hn)) = C (G)M , c c BR −δr − m˜ (φ)= φ(karn)e dn dr dνo(kX0 ). k∈K r∈R n∈N Proof. By definition,

BR (n−1)β + (o,π(u)) δβ − (o,π(u)) + − m˜ (φ)= φ(u)e u e u dmo(u )dνo(u )dt,

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where t = βu− (o, π(u)). Let u = karnX0. Then, since G − = MAN,wehave X0 − − − u = karnX0 = kX0 and

t = βu− (o, π(u)) = β − (o, karno)=β − (o, arno) kX0 X0 = limt→∞ d(o, a−to) − d(arno, a−to) = limt→∞ t − d(at+rna−t−r(at+ro),o) = limt→∞ t − d(at+ro, o)=−r.

δβ − (o,π(u)) − −δr − Therefore e u dνo(u )=e dνo(kX0 ). Also, by Lemma 7.1 for a ﬁxed g = kar and a variable z = n ∈ N,

− (n 1)β + (o,kar nπ(o)) + kar nX0 e dmo(karnX0 )=dλkar (n)=dλe(n)=dn.

Putting all of this together proves the claim.

Notation 7.4. (1) Let dk denote the probability Haar measure on K.Sincemo is Hn − a K-invariant probability measure on ∂ = K/M,wehavethatdk = dmo(kX0 ) + (and similarly dk = dmo(kX )). We ﬁx the Haar measure dg on G given as follows: −(n−1)r −1 for g = karn ∈ KAN, dg = e dn dr dk.SinceG is unimodular, dg = dg . (n−1)r Therefore if we express g = nark,thendg = e dn dr dk, and if we express g = arnk,thendg = dr dn dk. (2) For >0, let U denote the -neighborhood of e in G.Byanapproximate identity on G, we mean a family of non-negative continuous functions {ψ } >0 on ⊂ G with supp(ψ ) U and G ψ (g)dg =1. (3) For ξ ∈ C(M\K)andψ ∈ Cc(G) and a measurable Ω ⊂ K with MΩ=Ω, we deﬁne a function ξ ∗Ω ψ ∈ Cc(G/M)by

(7.4) ξ ∗Ω ψ(g):= ξ(k)ψ(gk) dk. k∈Ω

For ψ ∈ Cc(Γ\G), we deﬁne ξ ∗Ω ψ ∈ Cc(Γ\G/M) similarly.

Proposition 7.5. Let {ψ } >0 be an approximate identity on G.Letf ∈ C(M\K) ⊂ −1 − and Ω K be such that MΩ=Ωand νo(∂(Ω )X0 )=0.Then (7.5) lim m˜ BR(f ∗ ψ )= f(k−1) dν (kX−). → Ω o 0 0 k∈Ω−1

Proof. Note that for some uniform constants 1,2 > 0, we have for all k ∈ K and for all small >0,

−1 ⊂ −1 ⊂ ∩ ∩ −1 ∩ (7.6) k U U 1 k (A U 2 )(N U 2 )k (K U 2 ). Set K := (K ∩ U ), Ω =ΩK and Ω − = Ωk. 2 + k∈K In view of the decomposition G = ANK, for a function φ on K, we deﬁne a function Rφ on G by Rφ(g)=φ(k)forg = ank ∈ ANK. For any η>0, there exists >0 such that for all k ∈ K and g ∈ U ,

−1 −1 −1 R · (k ) − η ≤R · (k g) ≤R · (k )+η. f χΩ− f χΩ f χΩ+

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Now by Proposition 7.3, BR m˜ (f∗Ω ψ ) BR = g∈G k∈Ω ψ (gk)f(k ) dk dm˜ (g) −δr − = ψ (karnk )f(k )e dk dndrdνo(kX ) (k,ar,n )∈K×A×N k ∈Ω 0 −δr − = ψ (karnk )f(k )χΩ(k )e drdndk dνo(kX ) k∈K (ar,n,k )∈A×N×K 0 −δr − R · g = k∈K g∈Gψ (kg) f χΩ (g)e dgdνo(kX0 ), if g = arg nk δ −1 − ≤ 2 R · e k∈K g∈G ψ (g) f χΩ (k g) dgdνo(kX0 ), by (7.6) δ −1 − ≤ e 2 ψ (g)(R · (k )+η) dgdν (kX ) k∈K g∈G f χΩ+ o 0 δ −1 − = e 2 R · (k ) dν (kX )+η|ν | , as ψ (g) dg =1 k∈K f χΩ+ o 0 o G − − δ 2 − 1 | | = e k∈Ω 1 f(k ) dνo(kX0 )+η νo . + Since Ω = Ωandη>0 was arbitrarily chosen, >0 + BR ∗ ≤ −1 − lim sup m˜ (f Ω ψ ) f(k ) dνo(kX0 ). → ∈ −1 0 k Ω BR ∗ ≥ −1 − Similarly, lim inf →0 m˜ (f Ω ψ ) k∈int Ω−1 f(k ) dνo(kX0 ). Since we assume −1 − that νo(∂(Ω )X0 ) = 0, we obtain (7.5). 7.2. Setup for counting results. Until the end of this section, let V be a ﬁnite- dimensional vector space on which G acts linearly from the right and let w0 ∈ V .

We set H := Gw0 . 7.2.1. When H is a symmetric subgroup of G. Let H

7.2.2. When GRw0 is a parabolic subgroup of G. Suppose that GRw0 is a parabolic subgroup of G.Letθ be any Cartan involution of G and let K = Gθ.Then ∈ Hn G = GRw0 K.LetN be the unipotent radical of GRw0 .Leto be such that 1 n Go = K.ThenS˜ := N · o is a horosphere. Let E˜ ⊂ T (H ) be the unstable ˜ ˜ ∩ horosphere such that π(E)=S and let H =(GRw0 K)N. 7.2.3. Common structure in both cases. Let the notation be as in any of the above §§ ∈ 1 Hn ∩ ˜ ˜∗ · 7.2.1 or 7.2.2. Let X0 To( ) E and let E = H X0.IfH is symmetric and codim(S˜) > 1, or if it is the parabolic case, then E˜ is connected and E˜∗ = E˜. If H is symmetric and codim(S˜)=1,thenE˜ has two connected components, E˜+ − ∗ ∗ + containing X0 and E˜ containing −X0, and then either E˜ = E˜ or E˜ = E˜ .There exists a one-parameter subgroup A = {ar}⊂G consisting of R-diagonalizable Gr ∈ R elements such that (X0)=arX0 for all r .LetM = GX0 , which coincides ± with ZK (A), i.e., the centralizer of A in K,andA = {a±r : r ≥ 0}.LetN be the expanding horospherical subgroup with respect to {ar}. ∩ When GRw0 is parabolic, then Gw0 = MN = H where M = GRw0 K. Hence

N is the unipotent radical of GRw0 , so there is no conﬂict of notation. In the case when H is symmetric, E˜∗ = E˜ if and only if G = HA+K. In all cases, we have

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G = HAK.PutE = p(E˜), E∗ = p(E˜∗), and in the special cases when E˜ is not connected, we set E± = p(E˜±). ∗ ∼ 7.2.4. HAK decomposition of Haar measure on G. Note that E˜ =HX0 =H/(M∩H) and recall that − Leb (n 1)βv+ (o,π(v)) + dμE˜ (v)=e dmo(v ). ¯ There is a Haar measure dh on H such that for any ψ ∈ Cc(H), if we put ψ(h)= ∩ m∈M∩H ψ(hm) dm,wheredm denotes the probability Haar integral on M H, then ψ¯ ∈ C (H)M∩H = C (E˜), and c c ¯ Leb (7.7) ψdh= ψdμE˜ . H E˜ In view of the decompositions G = HA+K or G = HAK, there exists a function ρ : R → (0, ∞) such that we get the following Haar measure dg on G: For any ψ ∈ C (G), by [35, Theorem 8.1.1] c

(7.8) ψdg= ρ(r)ψ(hark) dhdrdk and G k∈K r∈R h∈H e(n−1)|r| if r →±∞and H is symmetric, (7.9) ρ(r) ∼ (n−1)r →±∞ e if r and GRw0 is parabolic, where R = {r ≥ 0} if G = HA+K,otherwiseR = R. In fact, the Haar measure dg described in Notation 7.4(1) and the Haar measure dg deﬁned in (7.8) are identical; see §8.

7.3. Extension of Theorem 1.8 to Γ\G for Zariski dense Γ. The result in this subsection will enable us to state our counting theorems for general norms, provided Γ is Zariski dense. Letm ¯ BR be the measure on Γ\G which is the M-invariant extension of mBR. That is, for ψ ∈ Cc(Γ\G), m¯ BR(ψ):=mBR(ψ¯), ¯ where ψ(p(gX0)) = m∈M ψ(Γgm) dm and dm denotes the Haar probability measure on M. As M normalizes N,¯mBR is invariant for the right-translation action of N on Γ\G. Theorem 7.6 (Flaminio-Spatzier [11, Cor. 1.6]). Suppose that Γ is Zariski dense and |mBMS| < ∞.Thenm¯ BR is N-ergodic. ˜ § Let H and E be as in 7.2.1 or 7.2.2 so that H = GE˜ .Letdh be the Haar measure on H deﬁned as in (7.7). By abuse of notation, we also denote by dh the measure on ΓH \H induced by dh. | PS| ∞ We recall that for Γ Zariski dense, μE < implies that the canonical map ΓH \H → Γ\G is proper by Theorem 2.21. Theorem 7.7. Let Γ be a Zariski dense discrete subgroup of G such that |mBMS| < ∞ | PS| ∞ ∈ \ and μE < . Then for any ψ Cc(Γ G), | PS| (n−1−δ)r μE BR lim e ψ(Γhar) dh = m¯ (ψ). r→∞ | BMS| h∈ΓH \H m

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Proof. Deﬁne a measure λr on Γ\G as follows: for any ψ ∈ Cc(Γ\G), (n−1−δ)r λr(ψ)=e ψ(Γhar) dh. h∈ΓH \H 1 ∼ Let q :Γ\G → T (Γ\Hn) = Γ\G/M be the natural quotient map. Then for ¯ 1 n ¯ ¯ any ψ ∈ Cc(T (Γ\H )), we have ψ(q(xmar)) = ψ(q(xar)) for any m ∈ M and x ∈ Γ\G,asM and A commute with each other, and hence ¯ ◦ (n−1−δ)r ¯ Leb q∗(λr)(ψ)=λr(ψ q)=e ψ(var) dμE (v). E

|μPS| → · BR E Therefore by Theorem 1.8, q∗(λr) C m ,whereC = |mBMS| . BR In order to show that λr weakly converges to Cm¯ , it suﬃces to show that BR every sequence λrk has a subsequence converging to Cm¯ . For any sequence rk →∞,sinceq is a proper map, after passing to a subsequence { } \ → ∈ of rk there exists a measure λ on Γ G such that λrk (φ) λ(φ) for every φ Cc(Γ\G). Therefore BR q∗(λ)=Cm . For any g ∈ G, deﬁne a measure gλ on Γ\G by gλ(A)=λ(Ag) for any measurable A ⊂ Γ\G.Nowforanyψ ∈ Cc(Γ\G), m∈M (mλ)(ψ) dm = m∈M Γ\G ψ(xm) dλ(x) dm (7.10) BR BR = q¯∗(λ)(ψ¯)=Cm (ψ¯)=Cm¯ (ψ). Claim 1. λ is N-invariant. → + Proof of Claim 1. Due to Lemma 2.1, the map h hX0 is a submersion, and hence there exists a neighborhood Ω of e in N and a continuous injective map → + + ∈ σ :Ω H such that σ(e)=e and σ(z)X0 = zX0 for all z Ω. ∈ Fix z Ω, let zk := ark za−rk , and let hk = σ(zk) for all large k.Thenbk = −1 − ∈ + → → →∞ z hk G = MAN . Therefore bk e and a−rk bkark e as k . k X0 Let ψ ∈ Cc(Γ\G). Given >0andx ∈ Γ\G,set

ψ +(x)= supψ(xg)andψ − =infψ(xg). g∈U g∈U

−1 −1 − Since ψ is uniformly continuous and ark z = hkbk ark = hkark (a rk bk ark ), we have for all large k and for all x ∈ Γ\G, ≤ ≤ ψ −(xhhkark ) ψ(xark z) ψ +(xhkark ). Since the measure dh is H-invariant, ≤ ψ(Γhark z) dh ψ +(Γhhkark ) dh = ψ +(Γhark ) dh. h∈ΓH \H ΓH \H ΓH \H

Similarly, we get a lower bound in terms of ψ −.Sinceλr → λ as k →∞, k

λ(ψ −) ≤ ψ(xz) dλ(x) ≤ λ(ψ +). Γ\G

Since ψ ∈ Cc(Γ\G), we have that λ(ψ ±) → λ(ψ)as → 0. Therefore the z-action preserves λ. This proves Claim 1.

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Claim 2. λ = Cm¯ BR. Proof of Claim 2. By (7.10), it is enough to show that λ is M-invariant. For any >0, deﬁne a measure η on Γ\G by 1 η := | | mλ dm, M m∈M where |M | = dm.ThensinceM normalizes N, η is N-invariant. By (7.10) M BR η m¯ . BR Therefore, sincem ¯ is N-ergodic by Theorem 7.6, there exists c > 0 such that BR BR η = c m¯ .Thusη is M-invariant, asm ¯ is M-invariant. If λ is not M-invariant, there exist ψ ∈ Cc(Γ\G), m0 ∈ M and β>0 such that λ(m0 · ψ) ≥ λ(ψ)+β.Thereexists>0 such that for all m ∈ M , λ((mm0)ψ) ≥ λ(m · ψ)+β/2. This implies that η (m0ψ) ≥ η (ψ)+β/2, which is a contradiction to the M-invariance of η . Hence Claim 2 is proved.

As noted before, this completes the proof of Theorem 7.7.

7.4. Statements of counting theorems. Now we describe the main counting results of this section. In the next two theorems, Theorems 7.8, and 7.10, we suppose that the following conditions hold for w0 ∈ V and Γ a non-elementary discrete torsion-free subgroup of G:

(1) w0Γ is discrete.

(2) H is a symmetric subgroup of G,orGRw0 is a parabolic subgroup of G. | BMS| ∞ | PS| ∞ (3) m < and μE < . Let λ ∈ N be the log of the largest eigenvalue of a1 on R-span(w0G), and set

λ w0ar −λ w0a−r (7.11) w0 := lim and w := lim . r→∞ eλr 0 r→∞ eλr Theorem 7.8 (Counting in sectors). Let · be a norm on V satisfying ±λ ±λ ∈ ∈ (7.12) w0 mk = w0 k , for all m M and k K,

and set BT := {v ∈ V : v 0, if E = Gw0 X0, = ± ⎩ μPS(E ) ± − − ∓ E λ 1 δ/λ ± δ·|mBMS| k∈K w0 k dνo(kX0 ) > 0, otherwise. λ Remark 7.9. (1) By [13, Lemma 4.2], we have w0 = 0. Also, if H is symmetric, −λ then w0 =0.

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(2) Since w0Γ is discrete, HΓ is closed in G, and hence ΓH is closed in G.It follows that the canonical imbedding (Γ ∩ H)\H → Γ\G is a proper injective map. This properness follows from a suitable open mapping theorem in the category of locally compact Hausdorﬀ second countable topological group actions. Therefore ∩ \ ˜ → \Hn ± the map (Γ GS˜) S Γ is a proper map. In particular, E and E are closed subsets of T1(Γ\Hn). (3) The condition (7.12) holds if · is K-invariant as inTheorem 1.2. There ∈ −1 ∈ R exists a Weyl group element k0 K such that k0 ark0 = a−r for all r .Then −λ λ · ±λ λ ∈ w0 = w0 k0. Therefore if is K-invariant, then w0 k = w0 for all k K. Then the limit (7.13) becomes (1.1). Thus Theorem 7.8 implies Theorem 1.2. (4) When Γ is Zariski dense in G, Theorem 7.8 holds for any norm on V without the condition (7.12) and for the Ω without the M-invariance condition. See §7.7 for details. ±λ ∩ ⊂ (5) Since w0 is ﬁxed by H ZK (A), if M = ZK (A) H, then condition (7.12) holds for any norm on V .WehaveM ⊂ H in the parabolic case. In the case when H is symmetric, if S˜ is a single point or S˜ is of codimension one, then M ⊂ H. Theorem 7.10 (Counting in cones). Suppose further that Γ is Zariski dense in G. Let Θ be a measurable subset of V and let { ∈ ±λ ∈ R+ } Ω± = k K : w0 k Θ . −1 ∓ · If νo(∂(Ω± X0 )) = 0, then for any norm on V , #(w Γ ∩ B ∩ R+Θ) 1 (7.14) lim 0 T = T →∞ T δ/λ δ ·|mBMS| ⎧ ⎨ PS λ −1 −δ/λ − − ˜ μE (E) k∈Ω 1 w0 k dνo(kX0 ), if E = HX0, × + ⎩ PS ± ±λ −1 −δ/λ ∓ μ (E ) −1 w k dν¯o(kX ), otherwise. E k∈Ω± 0 0

Note that if Γ is Zariski dense in G and if ∂(Ω±) is contained in a countable n union of proper real algebraic subvarieties of ∂H ,thenνo(∂(Ω±)) = 0 (see [11, Corollary 1.4] and [23, Remark 1.7(2)]). 7.5. Proof of the counting statements. We follow the counting technique of [9] and [10]. For a Borel subset Ω ⊂ K satisfying the condition of Theorem 7.8, we set + BT (Ω) = BT ∩ w0A Ω and deﬁne the following counting function on Γ\G:

FBT (Ω)(g):= χBT (Ω)(w0γg). ∈ \ γ Γw0 Γ We note that (7.15) F (e)=#(w Γ ∩ B (Ω)) = #(w Γ ∩ B ∩ (w A+Ω)). BT (Ω) 0 T 0 T 0 ∈ \ For ψ1,ψ2 Cc(Γ G), we set ψ1,ψ2 := Γ\G ψ1(g)ψ2(g) dg. Let ψ ∈ C (Γ\G). Then by (7.8), c FBT (Ω),ψ = χBT (Ω)(w0g)ψ(g) dg Γ \G w0

(7.16) = ψ(hark) dh ρ(r) drdk. ∈ { ≥ } ∈ \ k Ω r 0: w0ar k

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For any k ∈ K and T>0, deﬁne

(7.17) r(k, T)=sup{r>0:w0ark

− λr λ ≤ λ1r ∈ ≥ (7.18) w0ark e w0 k C1e , for all k K and r r1. − λ ≥ Put 0 =(λ λ1)/λ > 0andC2 =2C1/ infk∈K w0 k .LetT1 1besuch − 0 ≤ λ 1/λ ≥ r1 ≥ that C2T1 1/2and(1/2)(T1/ supk∈K w0 k ) e .ForT T1, we deﬁne functions r±(k, T)via − r±(k,T) λ 1/λ ± 0 (7.19) e =(T/ w0 k ) (1 C2T ). Then by elementary calculation using (7.18),

(7.20) r−(k, T) ≤ r(k, T) ≤ r+(k, T), for all T ≥ T1 and k ∈ K. By (7.12),

(7.21) r±(mk, T )=r±(k, T), for all m ∈ M and k ∈ K.

We note that by (7.19), given >0, for T1() suﬃciently large,

δr±(k,T) λ δ/λ ≥ (7.22) e =(1+O())(T/ w0 k ) for all T T1().

Proposition 7.11. For any non-negative ψ ∈ Cc(Γ\G), r−(k,T) r Leb ρ(r) ∗ ψ (G (v))dμ (v) drdk ≤F ,ψ k∈Ω 0 E k E BT (Ω) ≤ r+(k,T) Gr Leb k∈Ω 0 ρ(r) E∗ ψk( (v))dμE (v) drdk,

M ∼ 1 n where ψk ∈ Cc(Γ\G) = Cc(T (Γ\H )) is given by

ψk(g)= ψ(gmk)dm. m∈M Proof. By (7.7), (7.8), (7.16), (7.20), (7.21) and Lemma 7.1, we get FB T (Ω),ψ = k∈Ω {r≥0: w a k

Proposition 7.12. For any ψ ∈ Cc(Γ\G), we have PS ∗ −δ/λ μE (E ) BR lim T FB (Ω),ψ = · m (ξw ∗Ω ψ), T →∞ T δ ·|mBMS| 0 λ −δ/λ where ξw0 (k)= w0 k .

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Proof. Without loss of generality, we may assume that ψ is non-negative. For any >0andk ∈ K, by Theorem 1.8 and (7.9), there exists r0 > 0 such that for any r>r0, μPS(E∗) · mBR(ψ ) (7.23) e(n−1−δ)r ψ (Gr(v)) dμLeb(v)= E k + O(), k E | BMS| v∈E∗ m (7.24) ρ(r)=(1+O())e(n−1)r.

Since ψ ∈ Cc(Γ\G), the map K k → ψk is continuous with respect to the 1 n sup-norm on Cc(T (H )). Therefore, since K is compact, we can choose r0 > 0 independent of k ∈ K. Now for suﬃciently large T>1,

± r (k,T) r Leb ρ(r) ∗ ψk(G (v))dμ (v)dr r0 E E r±(k,T) (−n+1+δ)r (n−1−δ)r Gr Leb = ρ(r)e e ∗ ψk( (v)) dμE (v) dr (7.25) r0 E PS ∗ · BR μE (E ) m (ψk) r±(k,T) δr = BMS + O() (1 + O()) e dr |m | r0 μPS(E∗)·mBR(ψ ) T δ/λ wλk −δ/λ E k · 0 δ/λ δr0 = |mBMS| δ + O()T + O(e ), where the last equation follows from (7.22) for suﬃciently large T . 1 n Since E ⊂ T (Γ\H ) is a closed subset, ψ ∈ Cc(Γ\G)andK is compact, it follows that for ﬁxed r > 1, we have 0 Leb sup ψk(var) dμE (v)=O(1). |r|≤r0,k∈K E

Hence − Gr Leb (n 1)r0 (7.26) ρ(r) ψk( (v))dμE (v)dr = O(e ). {r: w0ar k 0 is arbitrary, we ﬁnish the proof.

Lemma 7.13 (Strong wavefront lemma). There exist >1 and 0 > 0 such that + for any 0 <<0 and g = hak ∈ HA K with a≥2,

gU ⊂ h(H ∩ U )a(A ∩ U )k(K ∩ U ), where g denotes the distance of g from e in G which is K-invariant. Proof. If H is symmetric, the result follows from [14, Theorem 4.1]. Now suppose that H = N is horospherical. We may assume that the distance from e in G is invariant under conjugation by elements of K.Letu ∈ U .Then −1 −1 kuk ∈ U .Writekuk = h1a1k1,whereh1 ∈ H ∩ U , a1 ∈ A ∩ U and k1 ∈ K ∩ U for some ≥ 1 independent of .Now −1 −1 −1 gu = haku = ha(kuk )k =(h(ah1a ))(aa1)k(k k1k). + −1 −1 Since a ∈ A and h1 ∈ H = N, by (7.1), ah1a ≤h1.Also,k k1k = k1. Hence gu has the required form.

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−1 Proof of Theorem 7.8(1). By the assumption that νo(∂(Ω )) = 0, for all suﬃ- ciently small >0, there exists an -neighborhood K of e in K such that for Ω =ΩK and Ω − = Ωk, + k∈K −1 − −1 (7.27) lim νo(Ω + Ω − )=0. →0 Let >1 as in Lemma 7.13. Then for T 1, − ⊂ ⊂ BT (Ω)U 1 B(1+ )T (Ω +)andB(1− )T (Ω −) BT (Ω)u. ∈ u U−1 Let ψ ∈ Cc(G) be a non-negative function supported on U −1 and ψ dg =1, and let Ψ ∈ Cc(Γ\G) be the Γ-average of ψ : (7.28) Ψ (g):= ψ (γg). γ∈Γ

≤ ≤ ∈ −1 Then FB(1−)T (Ω−)(g) FBT (Ω)(e) FB(1+)T (Ω+)(g) for all g U . Therefore, by integrating against Ψ ,wehave ≤ ≤ FB(1−)T (Ω−), Ψ FBT (Ω)(e) FB(1+)T (Ω+), Ψ .

Let ξw0 be as deﬁned in Proposition 7.12. By Proposition 7.5, for any η>0, there exists >0 such that BR ∗ BR ∗ −1 − m (ξw0 Ω Ψ )=m ˜ (ξw0 Ω ψ )= ξw0 (k ) dνo(kX0 )+O(η). k∈Ω−1 Therefore by Proposition 7.12,

−δ/λ →∞ · limT T FB(1±)T (Ω±), Ψ ∗ (7.29) μPS(E ) − − E · − 1 = ·| BMS| ∈ 1 ξw0 (k ) dνo(kX0 )+O(η). δ m k Ω± In view of (7.27), we get PS ∗ FBT (Ω)(e) μE (E ) −1 − lim = · ξw (k ) dνo(kX )+O(η). →∞ δ/λ ·| BMS| 0 0 T T δ m k∈Ω−1 Since η>0 is arbitrarily chosen, we ﬁnish the proof of (1).

+ Proposition 7.14. Suppose that H = Gw0 is symmetric and that G = HA K. ⊂ \ −1 + Let Ω M K such that νo(∂(Ω X0 )) = 0.Then

#(w Γ∩B ∩w A−Ω) →∞ 0 T 0 limT T δ/λ (7.30) − μPS(E ) − − − E λ 1 δ/λ + = δ·|mBMS| k∈Ω−1 w0 k dνo(kX0 ).

Proof. For k ∈ K and T>0, let s(k, T)=sup{r>0:w0a−rk 0andT0 > 0 such that if we deﬁne s±(k, T)via − − s±(k,T) ± 0 λ 1/λ e =(1 A0T )(T/ w0 k ) ,

then for all T ≥ T0,wehaves−(k, T) ≤ s(k, T) ≤ s+(k, T).

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1 n By Theorem 1.8, for any φ ∈ Cc(Γ\ T (H )), we have (n−1−δ)r G−r Leb limr→∞ e E+ φ( (v)) dμE (v) − − − μPS(E ) (n 1 δ)r ˜ Gr Leb E · BR ˜ = limr→∞ e E− φ( (v)) dμE (v)= δ·|mBMS| m (φ),

where φ˜(v)=φ(−v). − ∩ − Let BT (Ω) = BT w0A Ωand F − (g):= χ − (w0γg). BT (Ω) BT (Ω) ∈ \ γ Γw0 Γ In view of these observations, by arguing as in the proof of Proposition 7.12, we get that for any ψ ∈ Cc(Γ\G),

−δ/λ limT →∞ T F − ,ψ BT (Ω) −δ/λ = limT →∞ T ψ(ha−rk) dh ρ(r) drdk k∈Ω {r>0: w0a−r k

→ →−∞ Remark 7.15. If GRw0 is parabolic, then w0ar 0asr .Sincew0Γis discrete, − (7.31) #(w0Γ ∩ w0A K) < ∞. Proof of Theorem 7.8(2). If G = HA+K, then (2) follows from (1) by putting Ω=K. If H is symmetric and G = HA+K,thenG = HA+K HA−K, and then (2) follows by combining (1) and Proposition 7.14 and putting Ω = K. If GRw0 is parabolic, (7.13) follows from Theorem 7.8 and (7.31).

7.6. Counting in bisectors of HA+K coordinates. We state a counting result for bisectors in HA+K coordinates. For any g ∈ HA+K,weseta(g)tobethe + A -component of g, which is unique. Consider bounded Borel subsets Ω1 ⊂ H and Ω2 ⊂ K with Ω1(H ∩ M)=Ω1 and MΩ2 =Ω2.Set ∩ + NT (Ω1, Ω2)=#(Γ Ω1AT Ω2), + { ∈ + r } where AT = ar A : e

PS −1 − Theorem 7.16. If μE (∂(Ω1(X0))) = νo(∂(Ω2 (X0 ))) = 0,then

NT (Ω1, Ω2) 1 PS · −1 − lim = μE (Ω1(X0)) νo(Ω2 (X0 )). T →∞ T δ δ ·|mBMS| This result for H = K was also obtained by Roblin [31] by a diﬀerent approach. When Γ is a lattice in a semisimple Lie group G and H = K, the analogue of Theorem 7.16 was obtained in [12].

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Proof. We deﬁne the following function on Γ\G:

+ FT,Ω1,Ω2 (g):= χ (γg). Ω1AT Ω2 γ∈Γ

For ψ ∈ Cc(Γ\G), given >0, by Theorem 3.6 for suﬃciently large T>1, FT,Ω ,Ω ,ψ = + ψ(g)dg 1 2 g∈Ω1A Ω2 T

= r ψ(hark)ρ(ar)dhdrdk k∈Ω2 1≤e

= r ρ(ar) ψk(har)dh drdk + OT (1) k∈Ω2 T0≤e 0 there exist-neighborhoods H and K of e in H and K, respectively, such that for Ω − := Ω h, 1, h∈H(H∩M) 1 Ω + := Ω H (H ∩ M), Ω − := Ω k and Ω + := Ω K ,as → 0, 1, 1 2, k∈K 2 2, 2 PS −1 − −1 − − → → μE (Ω1, + (X0) Ω1, (X0)) 0,νo(Ω2, + (X0 ) Ω2, − (X0 )) 0.

By Lemma 7.13, for >1 as therein, there exists an -neighborhood U of G such that for all T 1,

+ + − ⊂ Ω1AT Ω2U 1 Ω1, + A(1+ )T Ω2, + , + + − − ⊂ Ω1, A(1− )T Ω2, g∈U − Ω1AT Ω2g. 1 Let ψ ∈ Cc(G) be a non-negative function supported on U −1 and ψ dg =1, and let Ψ ∈ C (Γ\G) be the Γ-average of ψ : c Ψ (g)= ψ (γg). γ∈Γ It follows that

(7.33) F − , Ψ ≤F (e) ≤F , Ψ . (1 )T,Ω1,− ,Ω2,− T,Ω1,Ω2 (1+ )T,Ω1,+ ,Ω2,+ On the other hand, by Proposition 7.5, BR ∗ −1 − lim m (χK Ω ± Ψ )=νo(Ω ± (X )). →0 2, 2, 0 Therefore by (7.32), PS −1 − μ (Ω ± (X ))ν (Ω ± (X )) −δ E 1, 0 o 2, 0 lim T F(1± )T,Ω ± ,Ω ± , Ψ = . T →∞ 1, 2, δ ·|mBMS| By (7.33) we get

FT,Ω1,Ω2 (e) 1 PS −1 − lim = μE (Ω1, + (X0))νo(Ω + (X0 )). T →∞ T δ δ ·|mBMS| 2,

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7.7. Counting theorems for Γ Zariski dense. InthecasewhenΓisZariski dense, Theorem 7.8 holds for any norm on V and for any Ω without the M- invariance condition. Similarly, Theorem 7.16 holds without the M-invariance assumption on Ω1 and Ω2. The reason that this generalization is possible is because for Γ Zariski dense, we use Theorem 7.7 instead of Theorem 1.8. In proving Theorem 7.8, the place where we needed the M-invariance of Ω is in Proposition 7.11. For general Ω, we replace this proposition by r−(k,T) k∈Ω 0 ρ(r) Γ \H ψk(har)dh drdk w0 ≤ ≤ r+(k,T) FBT (Ω),ψ ∈ ρ(r) \ ψk(har)dh drdk, k Ω 0 Γw0 H

where ψk(g):=ψ(gk) ∈ Cc(Γ\G) is simply the translation of ψ by k. Applying Theorem 7.7 to the inner integral in the above, we deduce in the same way as in the proof of Proposition 7.12 that for any ψ ∈ Cc(Γ\G), we have PS ∗ −δ/λ μE (E ) BR (7.34) lim T FB (Ω),ψ = · m¯ (ξw ∗Ω ψ), T →∞ T δ ·|mBMS| 0 BR § λ −δ/λ wherem ¯ is deﬁned as in 7.3 and ξw0 (k):= w0 k .Nowforageneral · norm on V , note that the function ξw0 (k) is not necessarily M-invariant. How- ever, for an approximate identity {ψ } >0 on G and any f ∈ C(K), the proof of Proposition 7.5 can be easily modiﬁed to prove (7.35) lim m¯ BR(f ∗ ψ )= f(k−1) dν (kX−). → Ω o 0 0 k∈Ω−1 → Hence applying (7.34) to ψ = ψ and (7.35) to f = ξw0 and by sending 0, we obtain PS ∗ −δ/λ μE (E ) λ −1 −δ/λ − (7.36) lim T · F (e)= · w k dνo(kX ). →∞ BT (Ω) ·| BMS| 0 0 T δ m k∈Ω−1 This explains the generalization of Theorem 7.8(1). The generalization for Theo- rem 7.8(2) and Theorem 7.16 can be done similarly.

Proof of Theorem 7.10. In view of the above explanation, the result can be deduced from Theorem 7.8 (or its combination with Proposition 7.14 or Remark 7.15) via elementary arguments; see [13].

8. Appendix: Equality of two Haar measures Let H be a symmetric group as in §7.2.1. As in Notation 7.4(1), consider the Haar measure on G corresponding to the Iwasawa decomposition G = NAK given by (n−1)t dg = e dn dt dq, for g = natq, n ∈ N, at ∈ A, q ∈ K. Corresponding to the generalized Cartan decomposition G = HAK,by(7.8)the Haar measure on G can be expressed as

dg = c0 · ρ(r) dh dr dk, for g = hark ∈ HAK,

where c0 > 0 is a constant. We note that dn is deﬁned by Lemma 7.1 and dh is determined by (7.7).

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Theorem 8.1. c0 =1.

− Proof. Let the notation be as in §7.1. Let N = {g ∈ G : a−rgar → e as r →∞}. − −r − Then for y ∈ Lie(N )wehavea−r exp(y)ar =exp(e y). In view of an NAN M decomposition of a small neighborhood of e in G,forh in such a neighborhood we write

h = n(x(h))ab(h)v(y(h))m(h), ∼ − ∼ − where x(h) ∈ Lie(N) = Rn 1 and n(x(h)) = exp(x(g)), y(h) ∈ Lie(N +) = Rn 1 and v(y(h)) = exp(y(h)), b(h) ∈ R and m(h) ∈ M.Inparticular,

+ + + (8.1) hX0 = n(x(h))ab(h)v(y(h))m(h)X0 = n(x(h))X0 In view of the decompositions G = HAK and G = NAK,forh ∈ H, r>0and k ∈ K,weexpress

hark = n(z(h, r, k))at(h,r,k)q(h, r, k), where q(h, r, k) ∈ K. Now for h in a small neighborhood of e in H,wehave

−r hark = n(x(h))ab(h)v(y(h))m(h)ark = n(x(h))ar+b(h)v(e y(h))(m(h)k). In view of a G = NAK decomposition,

−r v(e y(h)) = n(x1(h, r))ab1(h,r)k1(h, r), with −r max(x1(h, r), b1(h, r), k1(h, r))=O(e x(h)). Therefore,

hark = n(x(h))ar+b(h)n(x1(h, r))ab1(h,r)(k1(h, r)m(h)k)

= n(x(h)+x2(h, r))ar+b(h)+b1(h,r)(k1(h, r)m(h)k),

−r−b(h) where x2(h, r)=e x1(h, r). So

−2r (8.2) x2(h, r) = e O(x(h)). Therefore z(h, r, k)=n(x(h)+x (h, r)),t(h, r, k)=r + b(h)+b (h, r), (8.3) 2 1 q(h, r, k)=k1(h, r)m(h)k. Since z(h, r, k)=z(hm,r,e)andt(h, r, k)=t(hm,r,e) for any k ∈ K and m ∈ M ∩ H = G + ∩ H, we can write z(h, r, k)=z([h],r)andt([h],r,k)=t([h],r), X0 ∩ + where [h]=h(M H)=hX0 . Moreover, for any ﬁxed h and r,sincedk is K-invariant, we have that dq(h, r, k)=dk. For h in a small neighborhood of e in H, r>0andk ∈ K,

e(n−1)t(h,r,k) dn(z(h,r,k)) dt(h,r,k) dq(h,r,k) c0 = ρ(r) dh dr dk − e(n 1)t([h],r) dn(z([h],r)) dt([h],r) · dq(h,r,k) = ρ(r) dh dr dk e(n−1)t([h],r) dn(z([h],r)) dt([h],r) = ρ(r) dh dr ,

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because z and t do not depend on k and for ﬁxed (h, r)wehaved(q(h, r, k)) = dk. Now the numerator depends only on [h]=hM and m∈H∩M 1 dm = 1. Therefore,

(n−1)β + (o,n(z(h,r,k))o) (n−1)t([h],r) n(z([h],r))X e e 0 (8.4) c0 = − ρ(r)e(n 1)β[h](o,[h]o) dm (n(z([h],r))X+) dt([h],r) × o 0 . dmo([h]) dr

To compute c0, we evaluate the Radon-Nikodym derivative at the point ([h],r)= + ([e],s)=(X0 ,s) for any ﬁxed s>0. Then we consider the upper half space model Rn−1 × R Hn − ∞ + ∈ Rn−1 >0 for with o =(0, 1) and X0 = .ThenX0 =0 = n ∂H {∞}.Sincemo is equivalent to the Lebesgue measure, let dm (x) o + ∈ Rn−1 (8.5) 0

Fixing [h]=[e], we get z([e],r)=0,t([e],r)=r. Therefore ∂z2 (Φ1, Φ2)=(0, 1). Hence the Jacobian of Φ at ([h],r)=(0,s)is

| | J(Φ)(0,s)=∂z1 Φ1(0,s)

dm (n(x ([h],s)+x(h))X+) = o 2 0 at [h] = 0, by (8.3) dmo([h]) + dmo(n(x2([h],s)+x([h]))X0 ) = + , by (8.1) dmo(n(x([h]))X0 )

d(x2([h],s)+x([h])) = d(x([h])) at [h]=0=x([h]), by (8.5)

d(x2([h],s)) =1+ d(x([h])) at [h]=0=x([h]) =1+O(e−2s(n−1)), by (8.2).

Note that for a ﬁxed s, due to (8.1) and (8.5), x2([h],s) is a smooth function of x([h]). By (8.4), the Radon-Nikodym derivative at ([h],r)=([e],s)is

(n−1)β + (o,n(z([e],s))o) (n−1)t([e],s) n(z([e],s))X e e 0 · c0 = − J(Φ)(0,s) ρ(s)e(n 1)β[e](o,[e]o) =(e(n−1)s/ρ(s))(1 + O(e−2s(n−1))).

(n−1)s Since ρ(s)/e → 1ass →∞,wehavec0 =1.

Acknowledgements The authors thank Thomas Roblin for useful comments on an earlier version of this paper. The authors also thank the referee for carefully reading the paper and asking many pertinent questions, which led us to proving more general results and improving the overall presentation of the paper.

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Department of Mathematics, Brown University, Providence, Rhode Island 02912 and Korea Institute for Advanced Study, Seoul, Korea E-mail address: [email protected] Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 E-mail address: [email protected]

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