TORUS COUNTING AND SELF-JOININGS OF KLEINIAN GROUPS

. For any d ≥ 1, we obtain counting and equidistribution results for tori with small volume for a class of d -dimensional torus packings, invariant under a self-joining Γ ρ < (cid:81) di =1 PSL 2 ( C ) of a Kleinian group Γ formed by a d -tuple of convex-cocompact representations ρ = ( ρ 1 , · · · , ρ d ). More precisely, if P is a Γ ρ -admissible d -dimensional torus packing, then for any bounded subset E ⊂ C d with ∂E contained in a proper real algebraic subvariety, we have lim s → 0 s δ L 1 ( ρ ) · # { T ∈ P :

In this paper, we obtain counting and equidistribution results for a certain class of d-dimensional torus packings invariant under self-joinings of Kleinian groups for any d ≥ 1. One-dimensional torus packings are precisely circle packings.To motivate the formulation of our main results, we begin by reviewing counting results for circle packings that are invariant under Kleinian groups ( [15], [23], [24], [25], [27], etc).
Circle counting.A circle packing in the complex plane C is simply a nonempty family of circles in C, for which we allow intersections among themselves.In the whole paper, lines are also considered as circles of infinite radii.Let Γ < PSL 2 (C) = Isom + (H 3 ) be a Zariski-dense convex-cocompact discrete subgroup.We call a circle packing P Γ-admissible if • P consists of finitely many Γ-orbits of circles; • P is locally finite, in the sense that no infinite sequence of circles in P converges to a circle.We denote by 0 < δ Γ ≤ 2 the critical exponent of Γ i.e. the abscissa of convergence for the Poincare series P(s) := g∈Γ e −s d H 3 (gp,p) where p ∈ H 3 is any point and d H 3 is the hyperbolic metric so that (H 3 , d H 3 ) has constant curvature −1.The extended complex plane Ĉ = C ∪ {∞} can be regarded as the geometric boundary of H 3 .The limit set of Γ is the set of all accumulation points of the orbit Γ(z) of z ∈ Ĉ; we denote it by Λ Γ ⊂ Ĉ.
Theorem 1.1.[24] For any Γ-admissible circle packing P, there exists a constant c P > 0 such that for any bounded measurable subset E ⊂ C whose boundary is contained in a proper real algebraic subvariety of C, here ω Γ is the δ Γ -dimensional Hausdorff measure on C ∩ Λ Γ with respect to the Euclidean metric on C.This theorem holds for a more general class of circle packings invariant by geometrically finite Kleinian groups, which includes the famous Apollonian circle packings for which the relevant counting result was first obtained in [15] (see [24] for more details and examples).
Torus counting.The main goal of this paper is to prove a higher dimensional analogue of Theorem 1. Figure 1 shows some image of a 2-torus packing.Although the torus T = C 1 × C 2 in Fig. 1 appears to be in R 3 , it should be understood as a subset of R 4 , representing the Cartesian product of the boundary circles of two discs.We are interested in understanding the asymptotic counting and distribution of tori with small volumes in a torus packing that is invariant under a self-joining of a convex-cocompact Kleinian group.
Let Γ < PSL 2 (C) be a convex-cocompact discrete subgroup and ρ = (ρ 1 = id, ρ 2 , • • • , ρ d ) be a d-tuple of faithful convex-cocompact representations of Γ into PSL 2 (C).Let G = d i=1 PSL 2 (C).The self-joining of Γ via ρ is defined as the following discrete subgroup of G: Throughout the paper we will always assume that Γ ρ is Zariski-dense in G.Each ρ i induces a unique equivariant homeomorphism f i : Λ Γ → Λ ρ i (Γ) , which is called the ρ i -boundary map [36].In this paper, we define the limit set of Γ ρ by • ρ i (Γ)C i is a locally finite circle packing; . The second condition is equivalent to that is, the circular slice C 1 ∩Λ Γ completely determines the toric slice T ∩Λ ρ .Definition 1.3.A torus packing P is called Γ ρ -admissible if • P consists of finitely many Γ ρ -orbits of Γ ρ -admissible tori; • P is locally finite in the sense that no infinite sequence of tori in P converges to a torus.
Remark 1.4.We remark that when #(C 1 ∩ Λ Γ ) ≥ 3, the locally finiteness hypotheses in the above definition can be reduced to the local-finiteness of the circle packing ΓC 1 (see Prop. 3.11).
We denote by δ L 1 (ρ) the abscissa of convergence of the series for p ∈ H 3 , which is the critical exponent of Γ ρ with respect to the L 1 product metric on d i=1 (H 3 , d H 3 ).We first state the following special case of the main result of this paper.
Theorem 1.5.Let P be a Γ ρ -admissible torus packing.There exists a constant c P > 0 such that for any bounded measurable subset E ⊂ C d with boundary contained in a proper real algebraic subvariety, we have where ω Γρ is a locally finite Borel measure on C d ∩ Λ ρ which can be explicitly described.In particular, if P is bounded, then Remark 1.6.
(1) Since δ L 1 (ρ) is bounded above by the usual critical exponent δ Γρ of Γ ρ with respect to the Riemannian metric (which equals the L 2 product metric) on d i=1 H 3 , we have by [13,Coro. 3.6]; here the notation dim(•) means the Hausdorff dimension of a measurable subset of Ĉ ≃ S 2 with respect to the spherical metric.
, then for any bounded torus packing is a locally finite circle packing for all 1 ≤ i ≤ d.This is because the boundary map f i is the restriction to Λ ρ i (Γ) of the quasiconformal homeomorphism F i : Ĉ → Ĉ associated to ρ i , and under the hypothesis ∞ / ∈ ∪ d i=1 Λ ρ i (Γ) , the F i are bi-Hölder maps on any compact subset of C ( [7], [36]).
More general torus-counting theorems.In order to present a more general torus-counting theorem, we define the length vector of a torus T where we used the negative sign so that the i-th coordinate of v(T ) tends to +∞ as C i shrinks to a point.The following result is the main theorem of this paper.
Theorem 1.7.Let ψ be any linear form on R d such that ψ > 0 on (R ≥0 ) d − {0}.There exist δ ψ > 0 and a locally finite Borel measure ω ψ on Λ ρ ∩ C d depending only on Γ ρ and ψ for which the following hold: for any Γ ρadmissible torus packing P, there exists a constant c ψ = c P,ψ > 0 such that for any bounded measurable subset E ⊂ C d with boundary contained in a proper real algebraic subvariety, we have, as R → ∞, The description of the measure ω ψ (Def.6.1) depends on the higher rank Patterson-Sullivan theory.In fact, it is equivalent to the unique (Γ ρ , ψ 0 )conformal measure on Λ ρ , where ψ 0 is the unique Γ ρ -critical linear form (Def. 2.8) proportional to ψ.We refer to Def. 2.6 for the definition of δ ψ .Remark 1.9.
(1) Theorem 1.5 can be deduced from this theorem by considering the linear form ψ : (2) Our approach can also handle the case where ψ(v(T )) is replaced by the Euclidean norm of v(T ) in (1.8); indeed, the analysis involved in that case is easier due to the strict convexity of the Euclidean balls in R d (see the last subsection of Sec 8).(3) The fact that the sublevel sets {t ∈ R d : ψ(t) < c} are linear (hence not strictly convex) presents new technical difficulties which were not dealt with in related previous works such as [24] and [5].
We now discuss examples of admissible torus packings arising naturally from the Teichmüller theory of Kleinian groups.
(1) Let Γ < PSL 2 (C) be a Zariski-dense and convexcocompact subgroup whose domain of discontinuity Ω Γ := Ĉ − Λ Γ has a connected component which is a round open disk B. Let C 1 := ∂B and d ≥ 2. By the Teichmüller theory of Γ, which relates the Teichmüller space of the Riemann surface Γ\Ω Γ and the quasiconformal deformation space of Γ ([20, Thm.5.27], [19]) we may choose quasi-conformal deformations packing (see Figure 2 for an example when d = 2).Note also that P consists of disjoint tori, and hence gives rise to a genuine packing.(2) Let Γ be a rigid acylindrical convex-cocompact Kleinian group, that is, Ω Γ is a union of infinitely many round disks with mutually disjoint closures.Let ρ 0 : Γ → PSL 2 (C) be a quasiconformal deformation of Γ which is not a conjugation, and f : Ĉ → Ĉ the associated quasiconformal map.Denoting by C the space of all round circles in Ĉ, it follows from ( [21], [22], [2]) that the set of all circles C ∈ C such that #C ∩ Λ Γ ≥ 2 and f (C) is a circle is a finite union of closed Γ-orbits in C. Indeed, if C ∈ C meets Λ Γ at more than one point, then either C separates Λ Γ or C ⊂ Λ Γ .Since the set of circles contained in Λ Γ is a finite union of closed Γ-orbits, it suffices to note that the set of all separating circles such that f (C) is a circle is a finite union of closed Γ-orbits.This follows from [21, Thm.1.5] and [2, Thm.1.6], since otherwise such a set must be dense in the space C Λ Γ of all circles meeting Λ Γ , and hence f must map all circles in C Λ Γ to circles.That implies that f is conformal [20] and hence ρ is a conjugation, a contradiction.Therefore the following 2-dimensional torus packing On the proof of Theorem 1.7.First of all, the self-joining group Γ ρ is an Anosov subgroup of G introduced in [10] (see Def. (2.2)), which enables us to apply the general ergodic theory developed for Anosov subgroups.While certain types of counting problems for orbits of Anosov subgroups in affine symmetric spaces were studied in our earlier paper [5] using higher rank Patterson-Sullivan theory, there were certain serious technical restrictions imposed in [5] which made it unclear what kind of torus packing counting problems could be approached using techniques there.One of the main novelties of this paper is to have isolated a natural class of torus packings (which are provided by the Teichmüller theory of Kleinian groups) for which we can apply the counting machinery of [5].It is not hard to reduce the proof of Theorem 1.7 to the case where P is of the form Γ ρ T 0 , where T 0 is the product of the unit circles centered at the origin and ψ is a so-called Γ ρ -critical linear form (see Def. 2.8).As in [24], we first translate the counting problem for torus packings into an orbital counting problem in H\G where H = Stab G (T 0 ); by introducing a suitable bounded measurable subset B ψ (E, R) ⊂ H\G in (4.12), we are led to consider the asymptotic of as R → ∞.The key ingredient for obtaining (1.8) as R → ∞ is a description of the asymptotic behavior of for f ∈ C c (Γ ρ \G), as R tends to infinity, as given in Theorem 7.1.The Γ ρ -admissibility assumption on P = Γ ρ T 0 is used to guarantee • the existence of some compact subset S ⊂ Γ ρ ∩ H\H, independent of R, such that the integral (1.11) can be expressed as • the finiteness of the skinning constant of Γ ρ ∩ H\H (see (5.5)).
With this information, as well as the analysis of the asymptotic shape of the family of the subsets {B ψ (E, R) : R > 0}, we are able to apply the mixing result from [6,Thm. 3.4] and [4,Thm. 1.3 & Thm. 1.4], and the equidistribution result from [5] which describes the asymptotic of the integral (1.11) in terms of the Burger-Roblin measures introduced in [5].We emphasize that due to the higher rank nature of the subsets B ψ (E, R), combined with the linear nature of ψ, whose sublevel sets are not strictly concave, the uniformity aspect in these results (see Propositions 5.6 and 5.8 for the nature of the uniformity that is required) is crucial for our analysis.In fact, working on this article led us to conjecture the precise uniformity formulation of the mixing results in [4], which were verified and appeared in an updated version by the authors.Finally, we remark that the measure ω ψ is the leafwise measure of the Burger-Roblin measure on the strict upper triangular subgroup of G (≃ C d ) (see Proposition 6.3).

Organization.
• In Section 2, we start by recalling the basic higher rank Patterson-Sullivan theory of self-joining groups.• In Section 3, we discuss an important property of Γ ρ -admissible torus packings and its consequences.• In Section 4, we define the family {B ψ (E, R) ⊂ H\G : R > 0} and explain how Theorem 1.5 can be translated into an orbitalcounting problem for a Γ ρ -orbit in H\G with respect to the family {B ψ (E, R) : R > 0}.• In Section 5, mixing and and equidistribution results from [4] [5] will be recalled with an emphasis on their uniformity aspects.• In Section 6, the measure ω ψ will be given explicitly and analyzed.
• In Section 7, we prove the key technical ingredient (Theorem 7.1) of the paper, which accounts for the asymptotic distribution of the average of translates of the H-orbit over the set B ψ (E, R) as R → ∞. • In Section 8, we prove the main theorem (Theorem 1.5).
• In Section 9, we prove that every proper subvariety of C d has zero Patterson-Sullivan measure and hence zero ω ψ measure; this is shown for a general Anosov subgroup of a semisimple real algebraic group.
Acknowledgements.We would like to thank Dongryul Kim for useful conversations on a related topic.

Self-joinings and higher rank Patterson-Sullivan theory
Let H 3 = {(z, r) : z ∈ C, r > 0} denote the upper halfspace model of hyperbolic 3-space with constant curvature −1, d the hyperbolic metric on H 3 and o = (0, 1) ∈ H 3 .The geometric boundary of H 3 is the extended complex plane Ĉ := C ∪ {∞}, which is the Riemann sphere.The Möbius transformation action of the group PSL 2 (C) on Ĉ extends to the action on the compactification H 3 ∪ Ĉ, and gives rise to the identification PSL 2 (C) ≃ Isom • (H 3 ), the identity component of the isometry group of H 3 .Similarly, the product group acts on Ĉd component-wise, giving rise to an isomorphism of the group G with Isom • ( d i=1 H 3 ), the identity component of the isometry group of the Riemannian product (H 3 ) d .
Self-joinings of convex-cocompact subgroups.Let Γ < PSL 2 (C) be a torsion-free convex-cocompact subgroup, that is, the convex core of the associated hyperbolic manifold Γ\H 3 Definition 2.1.The self-joining of Γ by ρ is defined as the following discrete subgroup of G: Recall that throughout the entire paper we assume that Γ ρ is Zariski-dense in G.
Anosov subgroups.Let | • | denote the word length on Γ with respect to a fixed finite generating set.Since each ρ i is convex-cocompact , there exists In other words, Γ ρ is an Anosov subgroup (with respect to a minimal parabolic subgroup) (see [12] and [10]).This is the most important feature of the self-joining Γ ρ which will be used in this paper.We remark that any Anosov subgroup of G arises in this way in view of the characterization [12, Thm 1.5].
Limit set.The product F = Ĉd is equal to the Furstenberg boundary of G; note that for d > 1, F is not the geometric boundary of d i=1 H 3 .Let P < G be the product of the upper triangular subgroups of the PSL The limit set of Γ ρ in F is defined as the set of all accumulation points of any Γ ρ -orbits in d i=1 H 3 on F = Ĉd : This definition coincides with the definition of the limit set given by Benoist ([17, Lemma 2.13], [1]).Note that for d = 1, this is the usual limit set Λ Γ of the Kleinian group Γ.Let Λ ρ i (Γ) ⊂ Ĉ denote the usual limit set of ρ i (Γ).
By the convex-cocompact assumption on ρ i , there exists a unique ρ iequivariant homeomorphism (2.3) In particular, we have

.4)
We let We respectively identify R d and R d ≥0 with the Lie algebra a = log A and its positive Weyl chamber a + = log A + via the map t → log a t .For g = (g 1 , • • • , g d ) ∈ G, the Cartan projection of g is defined as Limit cone and its dual cone.Definition 2.5.
where ℓ i (g) denotes the length of the closed geodesic representing the conjugacy class of ρ i (g) Since sup g∈Γ (ℓ i (g)/ℓ j (g)) < ∞ for all i, j by the convex-cocompactness assumption, we have L ρ − {0} ⊂ int a + , where int C denotes the interior of a cone C. We denote by a * the space of all linear forms on a.The dual cone of L ρ is given by is defined as follows: Φ ρ (0) = 0 and for any vector u ∈ a + −{0}, where τ D is the abscissa of convergence of the series The following lemma is due to Quint.

Properties of admissible torus packings
Notations.We will be using the following notations throughout the paper: We also define the following subgroups: where We set note that M is equal to the centralizer of A in K.
Let C denote the space of all circles in Ĉ (recall that a union of line and {∞} is considered as a circle with infinite radius) and 2), we may consider a torus as an element of T , and a torus packing with a subset of T .
H-orbits corresponding to admissible torus packings.Throughout the paper, we fix the following torus where C 0 = {|z| = 1} is the unit circle centered at the origin.Note that Since G acts transitively on T , we can endow T ≃ G/H with the quotient topology on G/H.Similarly, the topology on C will be induced from PSL(2, C)/ PSU(1, 1).
We call a torus • P consists of finitely many Γ ρ -orbits of Γ ρ -admissible tori; • P is locally finite in the sense that no infinite sequence of tori in P converges to a torus.
Proof.Suppose not.Then there exist sequences and D be a Dirichlet fundamental domain for the action of Γ ∩ H 0 on the convex hull C 0 ⊂ H 3 of C 0 .By the admissibility hypothesis, ΓC 0 is a locally finite circle packing.Hence the inclusion map By replacing h i,1 with an element of (Γ ∩ H 0 )h i,1 and modifying g i if necessary, we may assume that By (3.6) and by the assumption that the sequence {s i,j It follows from the ρ j -equivariance of f j that ξ j = f j (ξ 1 ) for each 1 ≤ j ≤ d.We will need the following general fact from hyperbolic geometry: for any sequence h i ∈ H 0 and t i ≥ 0 (i ∈ N), the sequence accumulates on C 0 if and only if {h i ∈ H 0 : i ∈ N} is unbounded.In this case, (3.8) shares the same limit point with {h i o ∈ H 3 : i ∈ N} along any of its convergent subsequence.Now, since h i,ℓ → ∞, it follows from (3.7) and the above fact that ) by the assumption that P is Γ ρadmissible, we have ) and the previous fact from hyperbolic geometry, this implies that h i,1 is unbounded and To prove the proposition we will use the following lemma, which is equivalent to [17,Prop. 7.4] in view of the characterization of the limit cone L ρ as an asymptotic cone of {µ(γ) Lemma 3.10 (Uniform conicality of Λ ρ ).[17,Prop. 7.4] There exists a compact subset Q ⊂ G such that the following holds: for any g ∈ G with gP ∈ Λ ρ and any closed convex cone D ⊂ int a + ∪{0} whose interior contains L ρ − {0}, we can find sequences γ i ∈ Γ ρ and log a i → ∞ in D such that Proof of Proposition 3.9.Let Q ⊂ G be as in Lemma 3.10.Choose any closed convex cone D ⊂ int a + ∪{0} whose interior contains L ρ − {0}.Since the inclusion map Γ ρ ∩ H\H → Γ ρ \G is a proper map, Lemma 3.10 implies that By Proposition 3.4, the subset on the right-hand side is bounded.Therefore (1) follows.
Suppose (2) is false.Then there exists a bounded subset S ⊂ G and infinite sequences and Hγ i ̸ = Hγ j for i ̸ = j.Since the image of γ −1 i h i a t i = s −1 i ∈ S −1 under the projection G → Γ ρ \G is bounded, it follows again from Proposition 3.4 that there exists a sequence δ i ∈ Γ ρ ∩ H such that the sequence hi := δ i h i is bounded.Set γi := δ i γ i .Note that H γi = Hγ i and γi = hi a t i s i ∈ Γ ρ .Since both hi and s i are bounded, the sequences t i and µ(γ i ) are within bounded distance of each other.Now using the fact that L ρ is the asymptotic cone of {µ(γ) : γ ∈ Γ ρ }, and E ∩ L ρ = {0}, we have t i ̸ ∈ E for all sufficiently large i, which is a contradiction.
Closedness of Γ ρ T 0 .The following proposition says that local finiteness of We need to show that T ∈ Γ ρ T 0 .Since ΓC 0 is closed and hence locally finite by Lemma 3.3, we may assume that for all n ≥ 1, g n C 0 = C 1 by throwing away finitely many g n 's (recall and it contains at least 3 distinct points.Since two circles sharing three distinct points must be equal to each other, we get ρ i (g n )C 0 = C 1 for all 1 ≤ i ≤ d and all n.It follows that T n = T = T 0 for all n, proving the first claim.The second claim can be proved similarly.□ Although we won't be using the following proposition in the rest of our paper, it is of independent interest and extends the analogous fact for convexcocompact groups for d = 1.Proposition 3.12.Let T be a torus and H T be the stabilizer of Proof.Without loss of generality, we may assume that H T is the product of PSL is proper by Lemma 3.3 and ha n ∈ H T , it follows that there exists δn ∈ Γ ρ ∩ H T that δn ha n is bounded.This implies that ξ = h(H T ∩ P ) is a radial limit point of Γ ρ ∩ H T in H T /(H T ∩ P ).Hence we have shown that T ∩ Λ ρ is equal to the set Λ rad Γρ∩H T of all radial limit points of

Torus counting function for admissible torus packings
We write r(C) for the radius of a circle C. Given a torus We will call a linear form ψ ∈ a * positive if ψ > 0 on a + − {0}.
In the rest of this section, we fix • a Γ ρ -admissible torus packing P = Γ ρ T 0 ; • a positive Γ ρ -critical linear form ψ ∈ a * .Proof.By the local-finiteness of ρ i (Γ)C 0 , there are only finitely many circles in ρ i (Γ)C 0 of radius bounded from below intersecting a fixed bounded set.In particular, By the positivity hypothesis on ψ, we have c := inf v∈a + ,∥v∥=1 ψ(v) > 0 and hence ψ(v) ≥ c∥v∥ for all v ∈ a + .Hence where π i (E) denotes the projection of E to the i-th factor Ĉ. The last quantity is finite by the local-finiteness of ρ i (Γ)C 0 .This proves the claim.□ We will introduce a subset Bψ (E, R) ⊂ H\G and explain how N R (P, ψ, E) is related to the number of Γ ρ -orbits in the set Bψ (E, R).
Definition of Bψ (E, R).For R > 0, we define , where a t is defined as in (2.4).As ψ is positive, A + ψ,R is bounded.For any subset E ⊂ C d , we define where n z is defined as in (3.1).For any ε > 0, set The following proposition allows us to reformulate the counting problem in terms of the sets Bψ (E ± ε , R) (cf.[24, Proposition 3.7]): For ε > 0, set (4.7)The finiteness of q 0 (P, E, ε) can be seen as in the proof of Lemma 4.3.
We deduce the following.

Hence if a
for each i and hence ℓ ∈ K ε M by Lemma 4.13 (2).Since K ε M = M K ε and M ⊂ H, this proves the lemma.□ Further refinement.The following lemma appears in [24,Prop. 4.7] for the case d = 1, and this implies the general d-case as the computations can be reduced to each component.
Lemma 4.18.For any ε > 0 and a bounded subset Note that by Lemma 4.15 and Lemma 4.17 The claim now follows from the definition of Bψ (E, R). □ Corollary 4.21.For any ε > 0, there exist q 1 = q 1 (E, D, ε) > 0 and Proof.The first inequality is trivial.For the second inequality, choose a slightly smaller closed cone is a bounded set and hence applying Lemma 4.18 to the cone for some compact set Z ′ ⊂ H\G.Applying Proposition 3.9 with S = KN −E gives the desired conclusion. .□

Mixing and equidistribution with uniform bounds
We fix a positive Γ ρ -critical linear form ψ ∈ a * and the (Γ ρ , ψ)-PS measure ν ψ given by Lemma 2.10.In this section, we recall the results of [4] and [5] on mixing (Proposition 5.6) and equidistribution (Proposition 5.8), with emphasis placed on their uniformity aspects that are crucial in our application.
Burger-Roblin measures m BR and m BR * .Recall that denotes the product of upper triangular subgroups.We also denote by P = Stab G (0, • • • , 0) the product of lower triangular subgroups.
For g ∈ G, its visual images are defined by g + := gP ∈ F and g − := g P ∈ F.
Let F (2) denote the unique open G-orbit in F ×F under the diagonal action, that is, gives a homeomorphism G/M ≃ F (2) × a, called the Hopf parametrization of G/M .We define a locally finite Borel measure mBR ψ on G/M as follows: where m o is the unique K-invariant probability measure on F, db is the Lebesgue measure on a, and σ is the linear form on a defined by By abusing notation slightly, we will also use mBR ψ to denote the corresponding M -invariant measure on G induced by mBR ψ .The measure mBR ψ is left Γ ρ -invariant and induces an Ň -invariant locally finite measure on Γ ρ \G, which we denote by m BR ψ .
where dm, da, dň denote the Haar measures for M , A, Ň , respectively.
We note that in Lemma 5.4, dm is normalized to be a probability measure on M , da is normalized to be compatible with the restriction of the Killing form on the lie algebra of A, and dň is equivalently given by the density ň → e 2ρ(β ň+ (o,ňo)) dν 0 (ň + ) where ν 0 denotes the unique K-invariant probability measure on F.

Patterson-Sullivan measure µ PS
Γρ∩H\H,ψ [5,Definition 8.7].We define a measure µ PS H,ψ on H as follows: for ϕ ∈ C c (H), let where dp is a right-Haar probability measure on H ∩ P (note that H ∩ P is compact for the pair (G, H) we are considering); for h ∈ H/H ∩ P , h + is well-defined and independent of the choice of a representative.The measure defined above is Γ ρ ∩ H-invariant: H,ψ induces a locally finite Borel measure on Γ ρ \Γ ρ H ≃ Γ ρ ∩ H\H, which we denote by µ PS Γρ∩H\H,ψ .The skinning constant of Γ ρ ∩ H\H with respect to ν ψ is defined as the total mass: sk Γρ,ψ (H (5.5) Uniform mixing.We fix the unique unit vector u = u ψ ∈ int L ρ such that provided by Lemma 2.10.Since the cone a + is contained in the closed half space {ψ ≥ 0} and ψ(u) > 0, a + can be parameterized by the map The following mixing result is due to [6,Thm. 3.4] and [4, Thm.1.4 & Thm.1.5]: the uniform bound as stated in the second part is crucial in our application as remarked before.

The measure ω ψ
Fix a positive Γ ρ -critical linear form ψ ∈ a * and the (Γ ρ , ψ)-PS measure ν ψ given by Lemma 2.10.Definition 6.1.We define a locally finite Borel measure ω ψ = ω Γρ,ψ on C d as follows: for all f ∈ C c (C d ), For each small ε > 0, let ϕ ε ∈ C c (N ε A ε M ε Ňε ) be a non-negative function such that G ϕ ε dg = 1 where dg is a Haar measure on G and for any z where dm is a probability Haar measure on M .The main goal of this section is to establish Corollary 6.5, which roughly says Let E ⊂ C d be a fixed bounded Borel set and ε > 0 be small enough so that Recalling the definition of E ± ε from (4.4), we have On the other hand, if We now relate the Burger-Roblin measure of ϕ ε and Patterson-Sullivan measure of Φ ε .Proposition 6.3.There exist C, c > 0 such that for all sufficiently small ε > 0, we have Proof.By Lemma 5.4, we have where all the densities appearing in the expression are those of the corresponding Haar measures, except for dν ψ and dω ψ .Note that if for some c ′ ≥ 1 depending only on E. Decomposing the Haar measure dg on G according to AM Ň N and then restricting to with the implied constant depending only on E. Now using the maximum of ∥ Ad nz ∥ over z ∈ ±E together with the N AM Ň decomposition of exp Ad (6.4), for every z ′ ∈ C d we have giving the desired inequality.□ Combining Lemma 6.2 and Proposition 6.3 gives the following result.

Equidistribution in average.
We fix a positive Γ ρ -critical ψ ∈ a * , ν ψ and u = u ψ , continuing the notations from sections 4 and 5.We also fix a closed cone D ⊂ int a + such that int D ⊃ L ρ − {0} and set D := exp D as in (4.10) and (4.11).Recall the notation B ψ (E, R) = H\HKD ψ,R N −E for a bounded subset E ⊂ C d , and κ, ℓ > 0 given by Theorem 5.6.
The main goal of this section is to prove the following main technical ingredient of the proof of Theorem 1.7, using Proposition 5.8.Theorem 7.1.For any f ∈ C c (Γ ρ \G) and a bounded measurable subset where c Γρ,ψ := ker ψ e −ℓI(w) dw and In the above, d[g] denotes the G-invariant measure on H\G which is compatible to Haar measures dg and dh on G and H respectively, that is, for any Since ψ(w) = 0, we compute that for all R > 0, Q R (w) is an interval of the form Q R (w) = (0, 1 Φρ(u) R).The uniform bound in Proposition 5.8 enables us to use the dominated convergence theorem to prove the following result.
Proof.For simplicity, set c u = Φ ρ (u) in this proof.For all sufficiently large R > 0, we may rewrite p R (w) as where By Proposition 3.4, and using the expression of g with respect to the generalized Cartan decomposition G = HA + K, we can choose ϕ ∈ C c (Γ ρ ∩ H\H) depending only on the support of f and E such that for all z ∈ E. This will allow us to apply Proposition 5.8 directly to f H z .Furthermore, by Proposition 3.9(1), the support of µ PS Γρ∩H\H,ψ is compact, so we may additionally assume that ϕ = 1 on the support of µ PS Γρ∩H\H,ψ and hence sk Γρ,ψ (H) = µ PS Γρ∩H\H,ψ (ϕ).By Proposition 3.9(2), Since d[a t n z ] = e 2σ(t) dt dz where σ(t) = d i=1 t i , we deduce from Lemma 4.18 and the inclusion Since M ⊂ H, we have We now compute the upper limit in (7.5).Using (7.2) and (7.4) together with the fact that t = su + √ sw on a + hence dt = s Altogether, we have thus obtained Similarly, but applying Lemma 4.18 to Bψ (E − ℓε , R − ℓε) and Note that by Corollary 6.5, we have for all sufficiently small ε > 0. Since ω ψ (∂E) = 0, taking ε → 0 + completes the proof.

Proof of the main counting theorem
In this section, we prove the following main theorem of this paper.
Theorem 8.1.Let P be a Γ ρ -admissible torus packing.For any positive linear form ψ ∈ a * , there exist a constant c ψ = c P,ψ > 0 such that for any bounded measurable subset E ⊂ C d with boundary contained in a proper real algebraic subvariety, we have Since σ ∈ a * is positive, Theorem 1.5 is a special case of Theorem 8.1, with δ L 1 (ρ) = δ σ , c P = (2π) dδ c P,σ and ω ψ = ω Γρ,σ .
The proof of the following lemma is postponed until the final section (Theorem 9.2).Lemma 8.4.For any bounded measurable subset E ⊂ C d with ∂E contained in a proper real algebraic subvariety, we have ω ψ (∂E) = 0.
Since every homothety class of a positive linear form can be represented by a positive Γ ρ -critical linear form (Lemma 2.9) and δ ψ = 1 for critical linear forms, Theorem 8.1 follows from Lemma 8.4 and the following.for some constant c ψ > 0.
Special case: P = Γ ρ T 0 .We will first prove Proposition 8.5 for the special case when P = Γ ρ T 0 .This will allow us to apply the results obtained in previous sections.
Let D be as defined in (4.10), and for any R > 0, A R denote the Rneighborhood of e in A. Fix closed cones D ± ⊂ int a + such that Let D ± = exp D ± and R 0 > 0 be such that Recall the definitions of D ± ψ,R and E ± ε from (4.4) and (4.11).Now defining where D ± [R 0 ,R) = D ± ψ,R − A R 0 , we have the following inclusions.
Proof.Using unfolding, we have Since the set difference between Let ε 0 = (ℓ ′ + 2)ε.The above computation, combined with (8.9), (8.10) and Proposition 8.11 gives Corollary 6.5 now gives Since ω ψ (∂E) = 0 by Lemma 8.4, the regularity of ω ψ gives lim completing the proof.□ General case.Without loss of generality, we may assume that P consists of a single Γ ρ -orbit; hence let P = Γ ρ T be a Γ ρ -admissible torus packing.We write T = g 0 T 0 , where g 0 = n z 0 a t 0 ; here z 0 is the vector consisting of the centers of the circles of T and t 0 = log r 0 where r 0 = (r 1 , . . ., r d ) are the corresponding radii.Set Similarly to Proposition 4.8, we can obtain the following estimate of N R (P, ψ, E) in terms of Bψ (E ± ε , R).
Proposition 8.12.For any ε > 0, there exists q 0 = q 0 (P, ε) > 0 such that for any R > 0 and any bounded measurable subset E ⊂ C d , we have Note that Γ g 0 ρ is also a self-joing of convex cocompact representations.Let Λ g 0 ρ and L g 0 ρ denote its limit set and limit cone, respectively.It is immediate from the definition that Hence, we can apply the results obtained in previous sections for a new subgroup Γ g 0 ρ .

PS-measures are null on algebraic varieties
In this section, we prove that ν ψ (S) = 0 for any proper real algebraic subvariety S of Ĉd .Since ω ψ is absolutely continuous with respect to ν ψ , it follows that ω ψ (∂E) = 0 whenever E has boundary contained in a proper real algebraic subvariety, and in particular, Lemma 8.4 follows.
We will in fact prove this in a more general setup, which we now explain.Let G be any connected semisimple linear real algebraic group.Let P = M AN < G be a minimal parabolic subgroup with a fixed Langlands decomposition.Let a denote the Lie algebra of A. Let i denote the opposition involution on a.We remark that the opposition involution is non-trivial if and only if G has a simple factor of type A n (n ≥ 2), D 2n+1 (n ≥ 2) and E 6 [35, 1.5.1].For instance, when G is a product of rank one groups, i is trivial.

1 .
Let d ≥ 1.By a torus in C d we mean a Cartesian product of d-number of circles C 1 , • • • , C d ⊂ C.However, it will be convenient to consider it as a d-tuple of circlesT = (C 1 , • • • , C d ) (1.2) rather than a subset C 1 × • • • × C d ⊂ C d .A d-dimensional torus packing in C d issimply a nonempty family of d-tori in C d .The volume of T is given by Vol(T ) = d i=1 2π radius C i .

Figure 2 .
Figure 2. The left-hand side is the limit set of a convex-cocompact Kleinian group Γ and the right-hand side is the limit set of a quasi-conformal deformation, say, ρ 0 , of Γ. Denoting by f the associated quasiconformal map, f maps the first green circle, say C, to the second green circle.Hence the torus T = (C, f (C)) is a (id × ρ 0 )(Γ)-admissible torus.(image credit: Yongquan Zhang) denote the abscissa of convergence for the series s → γ∈Γρ e −sψ(µ(γ)) .Critical linear forms.Let ∥•∥ denote the Euclidean norm on a = R d .The growth indicator function Φ 26, Prop.5.1]  where ∂D := D ∩ C 0 ⊂ Ĉ denotes the boundary at infinity of D.

Proposition 8 . 5 .
For any positive Γ ρ -critical linear form ψ ∈ a * and any bounded measurable subset E ⊂ C d with ω ψ (∂E) = ∅, we have lim R→∞ e −R N R (P, ψ, E) = c ψ ω ψ (E) Theorem 7.1 then gives the claimed identity.□ Proof of Proposition 8.5 when P = Γ ρ T 0 .Note that the set difference between B 0 ψ (E, R) and B D,ψ (E, R) is bounded independent of R. Hence, by Proposition 4.8 and Corollary 4.21, lim inf R→∞