Rigidity, counting and equidistribution of quaternionic Cartan chains

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Rigidity, counting and equidistribution of quaternionic Cartan chains © 2022 the Authors Published version Parkkonen, Jouni; Paulin, Frédéric Parkkonen, J., & Paulin, F. (2022). Rigidity, counting and equidistribution of quaternionic Cartan chains. Annales Mathematiques Blaise Pascal, 28(1), 45-69. https://doi.org/10.5802/ambp.399 2022 ANNALES MATHÉMATIQUES BLAISE PASCAL Jouni Parkkonen & Frédéric Paulin Rigidity, counting and equidistribution of quaternionic Cartan chains Volume 28, no1 (2021), p. 45-69. <http://ambp.centre-mersenne.org/item?id=AMBP_2021__28_1_45_0> Cet article est mis à disposition selon les termes de la licence Creative Commons attribution 4.0. https://creativecommons.org/licenses/4.0/ L’accès aux articles de la revue « Annales mathématiques Blaise Pascal » (http://ambp.centre-mersenne.org/) , implique l’accord avec les conditions générales d’utilisation (http://ambp.centre-mersenne.org/legal/). Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS Clermont-Ferrand — France Publication membre du Centre Mersenne pour l’édition scientifique ouverte http://www.centre-mersenne.org/ Annales mathématiques Blaise Pascal 28, 45-69 (2021) Rigidity, counting and equidistribution of quaternionic Cartan chains Jouni Parkkonen Frédéric Paulin Abstract In this paper, we prove an analog of Cartan’s theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group. Rigidité, comptage et équidistribution de chaînes de Cartan quaternioniennes Résumé Dans ce papier, nous montrons un analogue d’un théorème de Cartan, disant que les transformations du bord des espaces hyperboliques quaternioniens qui préservent les chaînes sont des transformations projectives. Nous donnons un résultat de comptage et d’équidistribution pour les orbites de chaînes arithmétiques dans le groupe de Heisenberg quaternionien. 1. Introduction The sphere at infinity 𝜕∞𝑋 of a negatively curved symmetric space 𝑋 carries many rich structures, from the geometric, analytic and arithmetic points of view. When the sectional curvature is not constant, the possibilities are particularly rich, for instance with the Carnot–Carathéodory, sub-Riemannian or (hyper) CR structures (see for instance [ 4 , 10 , 12 , 14 , 17 ]), leading to strong rigidity properties, as Pansu’s rigidity theorem for quasi-isometries [ 18 ]. Arithmetic subgroups of the isometry group of 𝑋 endow the sphere at infinity of 𝑋 with arithmetic structures, and problems of equidistribution of rational points or subvarieties in 𝜕∞𝑋 , as well as in other homogeneous manifolds, have been intensively studied (see for instance [1, 2, 6, 8, 9, 11, 15, 22] and many others). In this paper, we study the quaternionic hyperbolic spaces 𝑋 , whose extreme rigidity is exemplified by the Margulis–Gromov–Schoen theorem in [ 13 ], proving, contrarily to the real or complex case, the arithmeticity of lattices in the isometry group of 𝑋 . As announced in [ 22 ], we prove a von Staudt–Cartan type of rigidity result for the family of all 3-sphere chains in the sphere at infinity of 𝑋 , and, analogously to the complex hyperbolic case treated in [ 20 ], an effective equidistribution result for the arithmetic Keywords: counting, equidistribution, Cartan chain, quaternionic Heisenberg group, Cygan distance, subRiemannian geometry, quaternionic hyperbolic geometry. 2020 Mathematics Subject Classification: 11E39, 11F06, 11N45, 20G20, 53C17, 53C55. 45 J. Parkkonen & F. Paulin chains in orbits of arithmetic groups built using maximal orders in rational quaternion algebras. More precisely, let H be Hamilton’s quaternion algebra over R , with 𝑥↦→ 𝑥 its conjugation, n : 𝑥↦→ 𝑥𝑥 its reduced norm, tr : 𝑥↦→ 𝑥+𝑥 its reduced trace. Let 𝑞 be the quaternionic Hermitian form on the right vector space H3over Hdefined by 𝑞(𝑧0, 𝑧1, 𝑧2)=−tr(𝑧0𝑧2) +n(𝑧1), and PU𝑞 its projective unitary group. It is the isometry group of the quaternionic hyperbolic plane H 2 H , realised as the negative cone of 𝑞 in the right projective plane P2 r(H) , and normalised to have maximal sectional curvature − 1. See Section 2 for a more complete description. The boundary at infinity 𝜕∞ H 2 H of H 2 H is the isotropic cone of 𝑞 in P2 r(H) , and the intersections with 𝜕∞ H 2 H of the quaternionic projective lines meeting H 2 H are called chains. We study them, giving their elementary properties and complete geometric descriptions in Section 3. Our first result is similar to Cartan’s theorem (see [ 7 , 10 ]) in the complex hyperbolic case. See Theorem 3.3 for a version in any dimension. Theorem 1.1. A chain-preserving transformation from the boundary at infinity of the quaternionic hyperbolic plane to itself is a projective unitary transformation. The boundary at infinity 𝜕∞ H 2 H of H 2 H , with the point ∞=[ 1:0:0 ] removed, identifies by the map (𝑤0, 𝑤) ↦→ [𝑤0:𝑤: 1]with the quaternionic Heisenberg group Heis7={(𝑤0, 𝑤) ∈ H×H:tr 𝑤0=n(𝑤)}, with group law (𝑤0, 𝑤)(𝑤0 0, 𝑤0)=(𝑤0+𝑤0 0+𝑤𝑤0, 𝑤 +𝑤0).(1.1) We endow the metabelian simply connected real Lie group Heis7 with its Cygan distance 𝑑Cyg , which is the unique left-invariant distance such that 𝑑Cyg((𝑤0, 𝑤),( 0 , 0 )) = ( 4 n(𝑤0))1 4 . The chains 𝐶 contained in Heis7 are ellipsoids, and have a natural center cen(𝐶)and radius (see Section 3). Let 𝐴 be a definite ( 𝐴⊗QR=H ) quaternion algebra over Q , with discriminant 𝐷𝐴 . Let O be a maximal order in 𝐴 . We refer for instance to [ 25 ] for background on quaternion algebras and orders. The group PU𝑞(O) of elements of PU𝑞 represented by matrices with coefficients in O is a (necessarily arithmetic) lattice in PU𝑞 . A chain 𝐶0 is said to be arithmetic over O if the orbit of some point of 𝐶0 under the stabiliser of 𝐶0 in PU𝑞(O) is dense in 𝐶0 . The stabiliser PU𝑞(O)∞ of [ 1:0:0 ] in PU𝑞(O) preserves the diameters of the chains for 𝑑Cyg . The following result (see Theorem 4.2 for an explicit and more general version) is an asymptotic counting result of the arithmetic chains in an orbit under the arithmetic group PU𝑞(O)when their Cygan diameter tends to 0. 46 Rigidity, counting and equidistribution Theorem 1.2. Let 𝐶0 be an arithmetic chain in 𝜕∞ H 2 H . There exists a constant 𝜅 > 0and an explicit constant 𝑐 > 0such that, as 𝜖→ 0, the number of chains modulo PU𝑞(O)∞ in the PU𝑞(O) -orbit of 𝐶0 , with Cygan diameter at least 𝜖 , is equal to 𝑐𝜖−10( 1 +O(𝜖𝜅)) . An arithmetic chain 𝐶0 bounds in H 2 H a homothetic copy of the real hyperbolic space of dimension 4. We denote by Covol(𝐶0) the volume of the quotient of this real hyperbolic space, normalised to have sectional curvature − 1, by the stabiliser PU𝑞(O)𝐶0 of 𝐶0 in PU𝑞(O) , and by 𝑚0 the order of the pointwise stabiliser of this real hyperbolic space in PU𝑞(O) . We endow the real Lie group Heis7 with its Haar measure HaarHeis7 normalised in such a way that the total mass of the induced measure on the quotient of Heis7 by its (uniform) lattice Heis7∩(O×O) is 𝐷2 𝐴 4 (see for instance [ 22 , Lem. 8 · 4] for an explanation of this normalisation). Let 𝑚𝐴= 72 if 𝐷𝐴 is even, and 𝑚𝐴= 1otherwise. Finally, we denote by Δ𝑥 the unit Dirac mass at any point 𝑥 . The following result proves that the centers of the arithmetic chains in an orbit under the arithmetic group PU𝑞(O) equidistribute in the quaternionic Heisenberg group. Theorem 1.3. For the weak-star convergence of measures on Heis7, we have 𝑚0𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1) 25515 224 Covol(𝐶0)𝜖10 ∑︁ [𝑔]∈PU𝑞(O)/PU𝑞(O)𝐶0 𝜖≤diam𝑑Cyg (𝑔𝐶0)<∞ Δcen(𝑔𝐶0) ∗ ⇀HaarHeis7. We refer to Section 4 for a version with congruences and error terms, and a more developped study of explicit examples of arithmetic chains. Acknowledgements The authors thank the snowy arctic conditions in Äkäslompolo in January 2020 which have provided an exceptional working environment. This research was supported by CNRS IEA BARP. The second author thanks the Laboratoire de Mathématiques Jean Leray at the Université de Nantes where this paper was completed. 2. Quaternionic hyperbolic spaces and Heisenberg groups In this section, we briefly recall some background on the quaternionic hyperbolic spaces and quaternionic Heisenberg groups, as mostly contained in [ 22 , §3 and §6], see also [ 16 , 23 ] (with different choices of quaternionic Hermitian form and normalisation of the curvature). 47 J. Parkkonen & F. Paulin Let H be Hamilton’s quaternion algebra over R , with 𝑥↦→ 𝑥 its conjugation, n : 𝑥↦→ 𝑥𝑥 its reduced norm, tr : 𝑥↦→ 𝑥+𝑥 its reduced trace and Im : 𝑥↦→ 1 2(𝑥−𝑥) its imaginary part map. We denote by ( 1 , 𝑖, 𝑗, 𝑘) the canonical basis of H as a real vector space, so that 𝑥0+𝑥1𝑖+𝑥2𝑗+𝑥3𝑘=𝑥0−𝑥1𝑖−𝑥2𝑗−𝑥3𝑘. Let Im H={𝑥∈H:tr 𝑥=0}=R𝑖+R𝑗+R𝑘 be the R -subspace of purely imaginary quaternions of H . For all 𝑤=(𝑤1, . . . , 𝑤𝑁) and 𝑤0=(𝑤0 1, . . . , 𝑤0 𝑁) intherightvector space H𝑁 over H ,we denoteby 𝑤·𝑤0=Í𝑁 𝑝=1𝑤𝑝𝑤0 𝑝 their standard quaternionic Hermitian product, and we define n(𝑤)=𝑤·𝑤=Í𝑁 𝑝=1n(𝑤𝑝) . We endow H𝑁with the standard Euclidean structure (𝑤, 𝑤0) ↦→ 1 2tr(𝑤·𝑤0). We fix 𝑛∈N−{ 0 , 1 } . On the right vector space H×H𝑛−1×H over H with coordinates (𝑧0, 𝑧, 𝑧𝑛), let 𝑞be the nondegenerate quaternionic Hermitian form 𝑞(𝑧0, 𝑧, 𝑧𝑛)=−tr(𝑧0𝑧𝑛) +n(𝑧)(2.1) of Witt signature (1, 𝑛), and let Φ:H𝑛+1×H𝑛+1→H, defined by Φ:(𝑧0, 𝑧, 𝑧𝑛),(𝑧0 0, 𝑧0, 𝑧0 𝑛)↦→ −𝑧0𝑧0 𝑛−𝑧𝑛𝑧0 0+𝑧·𝑧0,(2.2) be the associated quaternionic sesquilinear form. The Siegel domain model of the quaternionic hyperbolic 𝑛-space H𝑛 His (𝑤0, 𝑤) ∈ H×H𝑛−1:tr 𝑤0−n(𝑤)>0, endowed with the Riemannian metric d𝑠2 H𝑛 H =1 (tr 𝑤0−n(𝑤))2n(d𝑤0−d𝑤·𝑤)+(tr 𝑤0−n(𝑤))n(d𝑤). Its boundary at infinity is 𝜕∞H𝑛 H=(𝑤0, 𝑤) ∈ H×H𝑛−1:tr 𝑤0−n(𝑤)=0∪ {∞}. Aquaternionic geodesic line in H 𝑛 H is the image by an isometry of H 𝑛 H of the intersection of H 𝑛 H with the quaternionic line H× { 0 } . With our normalisation of the metric, a quaternionic geodesic line is a totally geodesic submanifold of real dimension 4and constant sectional curvature −4. The closed horoballs in H𝑛 Hcentred at ∞ ∈ 𝜕∞H𝑛 Hare the subsets H𝑠={(𝑤0, 𝑤) ∈ H𝑛 H:tr 𝑤0−n(𝑤) ≥ 𝑠},(2.3) and the horospheres centred at ∞ are their boundaries 𝜕H𝑠 , where 𝑠 ranges in ] 0 ,+∞[ . Note that, for every 𝑠∈ ]0,1], we have 𝑑(𝜕H1, 𝜕H𝑠)=−ln 𝑠 2.(2.4) 48 Rigidity, counting and equidistribution The Siegel domain H 𝑛 H embeds in the right quaternionic projective 𝑛 -space P𝑛 r(H) by the map (using homogeneous coordinates) (𝑤0, 𝑤) ↦→ [𝑤0:𝑤: 1]. By this map, we identify H 𝑛 H with its image, which when endowed with the isometric Riemannian metric, is called the projective model of H 𝑛 H . Note that this image is the negative cone of the quaternionic Hermitian form 𝑞 defined in Equation (2.1) : we have H 𝑛 H=[𝑧0 : 𝑧 : 𝑧𝑛] ∈ P𝑛 r(H) : 𝑞(𝑧0, 𝑧, 𝑧𝑛)< 0  . This embedding extends continuously to the boundary at infinity, by mapping the point (𝑤0, 𝑤) ∈ 𝜕∞ H 𝑛 H−{∞} to [𝑤0 : 𝑤 : 1 ] and ∞ to [ 1:0:0 ] , so that the image of 𝜕∞ H 𝑛 H is the isotropic cone of 𝑞 : we have 𝜕∞ H 𝑛 H=[𝑧0 : 𝑧 : 𝑧𝑛] ∈ P𝑛 r(H) : 𝑞(𝑧0, 𝑧, 𝑧𝑛)= 0  . A projective point [𝑧0:𝑧:𝑧𝑛] ∈ P𝑛 r(H)is positive if 𝑞(𝑧0, 𝑧, 𝑧𝑛)>0. For every 𝑁∈N, let 𝐼𝑁be the identity 𝑁×𝑁matrix. Let 𝐽=©« 0 0 −1 0𝐼𝑛−10 −1 0 0 ª®®¬ . The conjugate-transpose matrix of a quaternionic matrix 𝑋=(𝑥𝑝, 𝑝0)1≤𝑝≤𝑟,1≤𝑝0≤𝑠 in M𝑟,𝑠 (H)is 𝑋∗=(𝑥∗ 𝑝, 𝑝0=𝑥𝑝0, 𝑝)1≤𝑝≤𝑠,1≤𝑝0≤𝑟∈M𝑠,𝑟 (H). Let U𝑞={𝑔∈GL𝑛+1(H):𝑞◦𝑔=𝑞}={𝑔∈GL𝑛+1(H):𝑔∗𝐽𝑔 =𝐽} be the unitary group of 𝑞 . Its left linear action on H𝑛+1 induces a projective action on P𝑛 r(H)with kernel its center, which is reduced to {±𝐼𝑛+1}. The projective unitary group PU𝑞=U𝑞/{±𝐼𝑛+1} of 𝑞 acts faithfully on P𝑛 r(H) , preserving H 𝑛 H , and its restriction to H 𝑛 H is the full isometry group of H𝑛 H. A matrix 𝑋=©« 𝑎 𝛾∗𝑏 𝛼 𝑀 𝛽 𝑐 𝛿∗𝑑ª®®¬∈GL𝑛+1(H), 49 J. Parkkonen & F. Paulin with 𝑎, 𝑏, 𝑐, 𝑑 ∈H , 𝛼, 𝛽, 𝛾, 𝛿 ∈H𝑛−1 (identified with their column matrices in M𝑛−1,1(H) ) and 𝑀∈M𝑛−1,𝑛−1(H), belongs to U𝑞if and only if  𝑐𝑎 −𝛼∗𝛼+𝑎𝑐 =0 𝑑𝑏 −𝛽∗𝛽+𝑏𝑑 =0 −𝛿𝛾∗+𝑀∗𝑀−𝛾𝛿∗=𝐼𝑛−1 𝑑𝑎 −𝛽∗𝛼+𝑏𝑐 =1 𝛿𝑎 −𝑀∗𝛼+𝛾𝑐 =0 𝛿𝑏 −𝑀∗𝛽+𝛾𝑑 =0. (2.5) With Sp(𝑛− 1 )={𝑔∈GL𝑛+1(H) : 𝑔∗𝑔=𝐼𝑛−1} , an easy computation shows that the block upper triangular subgroup of U𝑞is B𝑞=©« 𝜇𝑟 𝜁∗1 2𝑟(n(𝜁) +𝑢)𝜇 0𝑈1 𝑟𝑈𝜁 𝜇 0 0 𝜇 𝑟ª®®¬ :𝜁∈H𝑛−1, 𝑢 ∈Im H, 𝑈∈Sp(𝑛−1), 𝜇 ∈Sp(1), 𝑟 > 0 . Its image PB𝑞=B𝑞/{±𝐼𝑛+1}in PU𝑞is equal to the stabiliser of ∞in PU𝑞. The quaternionic Heisenberg group Heis4𝑛−1 of dimension 4 𝑛− 1is the real Lie group structure on H𝑛−1×Im Hwith law (𝜁, 𝑢)(𝜁0, 𝑢0)=(𝜁+𝜁0, 𝑢 +𝑢0+2 Im 𝜁·𝜁0) and inverses (𝜁, 𝑢)−1=(−𝜁, −𝑢) . It identifies with the punctured boundary at infinity 𝜕∞H𝑛 H−{∞} by the map (𝜁, 𝑢) ↦→ (𝑤0, 𝑤)where (𝑤0, 𝑤)=n(𝜁) +𝑢 2, 𝜁hence (𝜁, 𝑢)=(𝑤, 2 Im 𝑤0),(2.6) and with a subgroup of PB𝑞⊂PU𝑞 , preserving every horoball H𝑠 for 𝑠 > 0, by the map (𝜁, 𝑢) ↦→ ±©« 1𝜁∗n(𝜁)+𝑢 2 0𝐼𝑛−1𝜁 0 0 1 ª®®¬ . Equation (2.6) allows to recover the definition of Heis7 given in the Introduction, for which the inverses are (𝑤0, 𝑤)−1=(−𝑤0+n(𝑤),−𝑤). For every (𝜁, 𝑢) ∈ Heis4𝑛−1 , the map (𝜁0, 𝑢0) ↦→ (𝜁, 𝑢)(𝜁0, 𝑢0) is the Heisenberg translation by (𝜁, 𝑢) . For every 𝜁∈H𝑛−1 , the Heisenberg translation by (𝜁, 0 ) is called a horizontal (Heisenberg) translation. For every 𝑢∈Im H , the Heisenberg translation by ( 0 , 𝑢) is called a vertical (Heisenberg) translation. The canonical map Π𝑣 : Heis4𝑛−1→H𝑛−1 defined by (𝜁, 𝑢) ↦→ 𝜁 is a real Lie group morphism, called the vertical projection, whose kernel is the center of Heis4𝑛−1 . For every 𝑈∈Sp(𝑛− 1 ) , 50 Rigidity, counting and equidistribution the map (𝜁, 𝑢) ↦→ (𝑈𝜁, 𝑢) is the Heisenberg rotation by 𝑈 . For every 𝜆 > 0, the map ℎ𝜆:(𝜁, 𝑢) ↦→ (𝜆𝜁, 𝜆2𝑢)is the Heisenberg dilation by 𝜆. The Cygan distance 𝑑Cyg on Heis4𝑛−1 is the unique left-invariant distance on the real Lie group Heis4𝑛−1such that 𝑑Cyg((𝜁, 𝑢),(0,0)) =n(𝜁)2+n(𝑢)1/4,(2.7) or equivalently 𝑑Cyg((𝑤0, 𝑤),( 0 , 0 )) =( 4 n(𝑤0))1 4 by Equation (2.6) . We introduce (see [ 19 , 20 ] in the complex case) the modified Cygan distance 𝑑00 Cyg , as the unique left-invariant map from Heis4𝑛−1×Heis4𝑛−1to [0,+∞[ such that 𝑑00 Cyg((𝜁, 𝑢),(0,0)) =(n(𝜁)2+n(𝑢))1/2 (n(𝜁)2+n(𝑢))1/2+n(𝜁)1/2,(2.8) or equivalently by Equation (2.6) 𝑑00 Cyg((𝑤0, 𝑤),(0,0)) =2n(𝑤0)1/2 (2n(𝑤0)1/2+n(𝑤))1/2. Though not actually a distance, the map 𝑑00 Cyg is symmetric and satisfies 1 √2𝑑Cyg ≤𝑑00 Cyg ≤𝑑Cyg. For every nonempty bounded subset 𝐸 of Heis4𝑛−1 , we define the diameter of 𝐸 for this almost distance as diam𝑑00 Cyg (𝐸)=sup 𝑥,𝑦 ∈𝐸 𝑑00 Cyg(𝑥, 𝑦). Note that the Cygan distance and the modified Cygan distance are invariant under Heisenberg translations and rotations, and that for every 𝜆 > 0, the Heisenberg dilation ℎ𝜆is a homothety of ratio 𝜆for both distances. Lemma 2.1. For every geodesic line ]𝑥, 𝑦[ in H 𝑛 H disjoint from the horoball H1 , the distance in H𝑛 Hbetween H1and ]𝑥, 𝑦[is equal to 𝑑(H1,]𝑥, 𝑦[) =−ln 1 √2𝑑00 Cyg(𝑥, 𝑦). Proof. By the invariance under Heisenberg translations of H1 , of the distance in H 𝑛 H and of the modified Cygan distance, we may assume that 𝑥=(𝑤0, 𝑤) ∈ 𝜕∞ H 𝑛 H−{∞,( 0 , 0 )} and 𝑦=( 0 , 0 ) ∈ 𝜕∞ H 𝑛 H− {∞} . By [ 22 , Lem. 6 · 4], the geodesic line from (𝑤0, 𝑤) to (0,0)is, up to translation at the source, the map 𝛾𝑤0,𝑤 :𝑡↦→ 𝑤0(1+𝑒2𝑡𝑤0)−1, 𝑤(1+𝑒2𝑡𝑤0)−1). 51 J. Parkkonen & F. Paulin H𝑛−1×Im H with its real tangent space at 𝑥 , the fiber 𝐸𝑥 of 𝐸 over 𝑥 is the horizontal subspace {(𝜁, 𝑢) ∈ H𝑛−1×Im H:𝑢=0}. Acalibration of 𝐸 is a 1-form 𝜔 on 𝑊 with values in Im H such that 𝐸=ker 𝜔 . Its Levi form is d𝜔. For instance, in the (𝜁, 𝑢)-coordinates of Heis4𝑛−1, the form 𝜔=d𝑢−2 Im(𝜁·d𝜁) is a calibration of 𝐸 (when restricted to 𝜕∞ H 𝑛 H−{∞} ). An easy computation shows that this calibration is invariant under Heisenberg translations and rotations: For every such transformation 𝛾 , we have 𝛾∗𝜔=𝜔 . The fact that 𝜔 is indeed a calibration follows by invariance since ker d 𝑢={(𝜁, 𝑢) ∈ H𝑛−1×Im H : 𝑢= 0 } . This calibration 𝜔 is scaled by the Heisenberg dilations as follows : for every 𝜆 > 0, we have (ℎ𝜆)∗𝜔=𝜆2𝜔. In the following result, we denote by 𝑣=𝑣1𝑖+𝑣2𝑗+𝑣3𝑘 the standard coordinate in Im H , and by d 𝑣 the tautological (Im H) -valued 1-form on Im H , so that for every 𝑥∈Im H , the map d 𝑣𝑥 : 𝑇𝑥Im H=Im H→Im H is the identity map. We denote by 𝜔1, 𝜔2, 𝜔3the standard coordinates of the calibration 𝜔, so that 𝜔=𝜔1𝑖+𝜔2𝑗+𝜔3𝑘. Given a chain 𝐶 in 𝜕 H 𝑛 H , let 𝜇=𝜇𝐶 be the (Borel positive) measure on Heis4𝑛−1 with support 𝐶∩Heis4𝑛−1 associated with the volume form 𝜔1∧𝜔2∧𝜔3 on 𝐶 . For instance, if 𝐶={(𝜁, 𝑢) ∈ H𝑛−1×Im H : 𝜁= 0 }∪{∞} is the standard vertical chain, then 𝜔|𝐶=d𝑢|𝐶, so that 𝜇𝐶is the (infinite) measure 𝜇𝐶=d𝑢1d𝑢2d𝑢3, whose restriction to the Euclidean space 𝐶−{∞} ={ 0 }×Im H is the standard Lebesgue measure. Given a nonzero measure 𝜇 with compact support on a finite dimensional real affine space 𝑉, the barycenter (or centroid) of 𝜇is the point bar(𝜇)of 𝑉defined by bar(𝜇)=1 𝜇(𝑉)∫𝑥∈𝑉 𝑥d𝜇(𝑥). For instance, when 𝜇 is supported on a finite set 𝑆 , then bar(𝜇) is the usual affine barycenter of the weighted family of points 𝑠, 𝜇({𝑠}) 𝜇(𝑆)𝑠∈𝑆. We denote the open ball of center 0and radius 𝑟 in the Euclidean space Im H by 𝐵(𝑟) . Recall that the radius of a finite chain 𝐶is denoted by 𝑅𝐶. Proposition 3.4. Let 𝐶be a chain in 𝜕∞H𝑛 Hand 𝑐∈𝐶. (1) If 𝐶 is a finite chain, then the center of the chain 𝐶 is equal to the barycenter of the measure 𝜇𝐶: cen(𝐶)=bar(𝜇𝐶). 58 Rigidity, counting and equidistribution (2) If 𝐶 is a vertical chain, there is a diffeomorphism 𝜏=𝜏𝐶 : Im H→𝐶−{∞} such that 𝜏∗𝜔= d 𝑣 , unique up to postcomposition by a vertical Heisenberg translation. For every Heisenberg translation or rotation 𝛾, we have 𝜏𝛾𝐶 =𝛾◦𝜏𝐶. (3) If 𝐶 is a finite chain, there exists a smooth diffeomorphism 𝜏=𝜏𝐶,𝑐 from 𝐵( 2 𝜋𝑅2 𝐶) to 𝐶− {𝑐} , admitting a continuous extension to 𝜕𝐵( 2 𝜋𝑅2 𝐶) sending this sphere to 𝑐 , such that 𝜏∗𝜔= d 𝑣 . This mapping is unique up to postcomposition by a Heisenberg rotation preserving 𝐶 and 𝑐 , and 2 𝜋𝑅2 𝐶 is the unique radius for which such a mapping exists. For every Heisenberg translation or rotation 𝛾, we have 𝜏𝛾𝐶,𝛾𝑐 =𝛾◦𝜏𝐶,𝑐. Proof. (1). Note that Heis4𝑛−1=H𝑛−1×Im H has a natural structure of a real affine space, and that the elements of PB𝑞 act by affine transformations on Heis4𝑛−1 . This can be seen for instance by saying that Heis4𝑛−1 , identified with the boundary of the projective model of H 𝑛 H minus {∞} , is a PB𝑞 -invariant affine subspace of the affine chart of the quaternionic projective space defined by the quaternionic projective hyperplane [𝑧0 : 𝑧 : 𝑧𝑛] ∈ P𝑛 r(H) : 𝑧𝑛= 0  , and that the quaternionic projective transformations preserving this hyperplane act by affine transformations on the associated affine chart. Another way is to check, by an easy computation, that the Heisenberg translations, rotations and dilations preserve the barycenters in the real affine space H𝑛−1×Im H : For instance, for all (𝜁0, 𝑢0),(𝜁, 𝑢),(𝜁0, 𝑢0) ∈ Heis4𝑛−1and 𝑡∈ [0,1], we have (𝜁0, 𝑢0) · 𝑡(𝜁, 𝑢)+(1−𝑡)(𝜁0, 𝑢0)=𝑡(𝜁0, 𝑢0)·(𝜁, 𝑢)+(1−𝑡)(𝜁0, 𝑢0)·(𝜁0, 𝑢0). In particular, the barycenters of measures 𝜇 with compact support on Heis4𝑛−1 are equivariant under the Heisenberg translations, rotations and dilations: For every such transformation 𝛾, we have bar(𝛾∗𝜇)=𝛾bar(𝜇).(3.3) In order to prove Assertion (1) , by Equations (3.2) and (3.3) , and by the transitivity properties of the Heisenberg translations and dilations on chains, we may assume that 𝑛= 2and that 𝐶 is a Euclidean sphere with center ( 0 , 0 ) and radius 1in the horizontal subspace {(𝜁, 𝑢) ∈ H𝑛−1×H : 𝑢= 0 } . Since the Im H -valued 1-form 𝜔|𝐶 is invariant under the Heisenberg rotations, the volume form 𝜔1∧𝜔2∧𝜔3 on 𝐶 is invariant under the Heisenberg rotations. Since the only measure on 𝐶 invariant under the Heisenberg rotations is, up to a scalar multiple, the Lebesgue measure on the Euclidean sphere 𝐶 , the measure 𝜇𝐶 is a multiple of the Lebesgue measure on 𝐶 . This can also be proved by a direct computation: On the Euclidean sphere 𝐶, with 𝜁=𝜁0+𝜁1𝑖+𝜁2𝑗+𝜁3𝑘, we have 𝜔1∧𝜔2∧𝜔3=−8 3 ∑︁ 𝑖=0(−1)𝑖𝜁𝑖d𝜁0∧···∧ c d𝜁𝑖∧···∧d𝜁3. 59 J. Parkkonen & F. Paulin Since the barycenter of this measure is exactly the origin ( 0 , 0 ) , which is the center of the finite chain 𝐶, this proves Assertion (1). (2). First assume that 𝐶is the standard vertical chain 𝐶∞={(𝜁, 𝑢) ∈ H𝑛−1×Im H:𝜁=0} ∪{∞}. Let 𝜏=𝜏𝐶∞ : 𝑣↦→ ( 0 , 𝑣) . Then 𝜏 is a diffeomorphism from Im H onto 𝐶∞−{∞} , such that 𝜏∗( d 𝑢− 2 Im(𝜁 d 𝜁)) = d 𝑣 . For every vertical Heisenberg translation 𝛾 , the map 𝛾◦𝜏 is also a diffeomorphism from Im H onto 𝐶∞−{∞} , and since 𝜔 is invariant under the Heisenberg translations, we also have (𝛾◦𝜏)∗𝜔=d𝑣. If 𝜎 : Im H→𝐶∞− {∞} is another diffeomorphism such that 𝜎∗𝜔= d 𝑣 , then for every 𝑣∈Im H , we have 𝜎0(𝑣)−𝜏0(𝑣) ∈ 𝑇𝐶∞∩ker 𝜔={ 0 } , thus the maps 𝜎 and 𝜏 differ by an element of the vector subspace 𝐶∞ . Therefore there exists a vertical Heisenberg translation 𝛾such that 𝜎=𝛾◦𝜏. Now, if 𝐶 is another vertical chain, there exists a composition 𝛾 of Heisenberg translations and rotations such that 𝐶=𝛾𝐶∞ . Defining 𝜏𝐶=𝛾◦𝜏𝐶∞ gives a diffeomorphism from Im H onto 𝐶−{∞} such that 𝜏𝐶∗𝜔= d 𝑣 , by the invariance of 𝜔 under the Heisenberg translations and rotations. This proves Assertion (2). (3). First assume that 𝐶is the Euclidean 3-sphere (𝜁, 𝑢) ∈ H𝑛−1×Im H:n(𝜁1)=𝑅2and 𝑢=𝜁2=··· =𝜁𝑛−1=0, and that 𝑐=(𝜁𝑐=(−𝑅, 0 , . . . , 0 ), 𝑢𝑐= 0 ) . Note that 𝑅 is the radius of the finite chain 𝐶 . By the properties of the exponential map of the Lie group of unit quaternions, whose tangent space at the identity element 1is Im H, the smooth map 𝜏=𝜏𝐶,𝑐 :𝑣↦→ 𝜁=(𝑅𝑒−𝑣/(2𝑅2),0, . . . , 0), 𝑢 =0 from Im H to 𝐶 is a diffeomorphism from 𝐵( 2 𝜋𝑅2) onto 𝐶−{𝑐} . It extends continuously (and even smoothly) to the sphere 𝜕𝐵( 2 𝜋𝑅2) , mapping this sphere to 𝑐 . Considering 𝜁 as a function of 𝑣 , we have d 𝜁=(− 1 2𝑅𝑒−𝑣/(2𝑅2) d 𝑣, 0 , . . . , 0 ) . Hence, since 𝑣 and d 𝑣 are purely imaginary quaternions, we have 𝜏∗𝜔=−2 Im(𝜁·d𝜁)=−2 Im𝑅𝑒−¯𝑣/(2𝑅2)−1 2𝑅𝑒−𝑣/(2𝑅2)d𝑣=d𝑣. The uniqueness of 𝜏 up to postcomposition by a Heisenberg rotation preserving 𝐶 and 𝑐 , and the extension to the other chains, follow as previously from the fact that the chains are transverse to the quaternionic contact structure on Heis4𝑛−1and by invariance of the calibration 𝜔under the Heisenberg translations and rotations.  60 Rigidity, counting and equidistribution 4. Counting and equidistribution of arithmetic chains in hyperspherical geometry In this section, we prove (generalised versions of) Theorems 1.2 and 1.3 of the introduction. We start by recalling a general statement, coming from a special case of the main results of [21], that has been made explicit in [22]. Let Γ be a lattice in PU𝑞 . Let 𝐷− and 𝐷+ be nonempty proper closed convex subsets of H 𝑛 H , with stabilisers Γ𝐷− and Γ𝐷+ in Γ respectively, such that the families (𝛾𝐷−)𝛾∈Γ/Γ𝐷− and (𝛾𝐷+)𝛾∈Γ/Γ𝐷+ are locally finite in H 𝑛 H . For all 𝛾, 𝛾0 in Γ , the convex sets 𝛾𝐷− and 𝛾0𝐷+ have a common perpendicular if and only if their closures 𝛾𝐷− and 𝛾0𝐷+ in H 𝑛 H∪𝜕∞ H 𝑛 H do not intersect. We denote by 𝛼𝛾,𝛾0 this common perpendicular, starting from 𝛾𝐷−at time 𝑡=0, and by ℓ(𝛼𝛾,𝛾0)its length. The multiplicity of 𝛼𝛾,𝛾0is 𝑚𝛾,𝛾0=1 card(𝛾Γ𝐷−𝛾−1∩𝛾0Γ𝐷+𝛾0−1), which equals 1for all 𝛾, 𝛾0∈Γ when Γ acts freely on 𝑇1 H 𝑛 H (for instance when Γ is torsion-free). For all 𝑠 > 0and 𝑥∈𝜕𝐷−, let 𝑚𝑠(𝑥)=∑︁ 𝛾∈Γ/Γ𝐷+:𝐷−∩𝛾𝐷+=∅, 𝛼𝑒,𝛾 (0)=𝑥, ℓ (𝛼𝑒,𝛾)≤𝑠 𝑚𝑒,𝛾 be the multiplicity of 𝑥 as the origin of common perpendiculars with length at most 𝑠 from 𝐷−to the elements of the Γ-orbit of 𝐷+. For every 𝑠 > 0, let N𝐷−,𝐷+(𝑠)=∑︁ (𝛾,𝛾0)∈Γ\((Γ/Γ𝐷−)×(Γ/Γ𝐷+)) :𝛾𝐷−∩𝛾0𝐷+=∅, ℓ (𝛼𝛾,𝛾0)≤𝑠 𝑚𝛾,𝛾0, where Γ acts diagonally on Γ×Γ . When Γ has no torsion, N𝐷−,𝐷+(𝑠) is the number (with multiplicities coming from the fact that Γ𝐷±\𝐷± is not assumed to be embedded in Γ\ H 𝑛 H ) of the common perpendiculars of length at most 𝑠 between the images of 𝐷− and 𝐷+in Γ\H𝑛 H. The following statement is a special case of [ 22 , Thm. 8 · 1]. We denote by Δ𝑥 the unit Dirac mass at a point 𝑥. Theorem 4.1. Let 𝐷− be a horoball in H 𝑛 H centred at a parabolic fixed point of Γ and let 𝐷+ be a quaternionic geodesic line in H 𝑛 H such that Γ𝐷+\𝐷+ has finite volume. Let 𝑚+ be the order of the pointwise stabiliser of 𝐷+in Γand let 𝑐(𝐷−, 𝐷+)=2(𝑛−1)(2𝑛−1) 𝜋2𝑚+ Vol(Γ𝐷−\𝐷−)Vol(Γ𝐷+\𝐷+) Vol(Γ\H𝑛 H). 61 J. Parkkonen & F. Paulin There exists 𝜅 > 0such that, as 𝑠→ +∞, N𝐷−,𝐷+(𝑠)=𝑐(𝐷−, 𝐷+)𝑒(4𝑛+2)𝑠1+O(𝑒−𝜅𝑠). Furthermore, the origins of the common perpendiculars from 𝐷− to the images of 𝐷+ under the elements of Γ equidistribute in 𝜕𝐷− to the induced Riemannian measure: As 𝑠→ +∞, we have 2(2𝑛+1)Vol(Γ𝐷−\𝐷−) 𝑐(𝐷−, 𝐷+)𝑒−(4𝑛+2)𝑠∑︁ 𝑥∈𝜕𝐷− 𝑚𝑠(𝑥)Δ𝑥∗ ⇀vol𝜕𝐷−.(4.1) For smooth functions 𝜓 with compact support on 𝜕𝐷− , there is an error term in the equidistribution claim of Theorem 4.1 when the measures on both sides are evaluated on 𝜓 , of the form O(𝑒−𝜅𝑠 k𝜓kℓ) where 𝜅 > 0and k𝜓kℓ is the Sobolev norm of 𝜓 for some ℓ∈N. From now on, we assume that 𝑛= 2. Let 𝐴 , 𝐷𝐴 , 𝑚𝐴 and O be as in the Introduction. We denote by |O×| the order of the unit group of O , equal to 24 if 𝐷𝐴= 2, to 12 if 𝐷𝐴= 3, or else to 2,4or 6. See for instance [ 25 ]. As usual, by Î𝑝|𝐷𝐴 , we mean a product where 𝑝ranges over the prime positive numbers dividing 𝐷𝐴. For every chain 𝐶 in 𝜕∞ H 2 H , let 𝐿𝐶 be the quaternionic projective line in P2 r(H) such that 𝐶=𝐿𝐶∩𝜕∞ H 2 H , and let 𝐷𝐶=𝐿𝐶∩ H 2 H be the associated quaternionic geodesic line. For every finite index subgroup 𝐺 of the arithmetic lattice PU𝑞(O) , we denote by 𝐺𝐶 the stabiliser of 𝐶 in 𝐺 , by 𝐺∞ the stabiliser of ∞ in 𝐺 , and by Covol𝐺(𝐶) the volume of the orbifold 𝐺𝐶\𝐷𝐶 for the Riemannian metric of constant sectional curvature − 1on the real hyperbolic 4-space 𝐷𝐶 . Recall that a chain 𝐶 is arithmetic over O if and only if the stabiliser in PU𝑞(O) (or equivalently in 𝐺 ) of the quaternionic geodesic line 𝐷𝐶 has finite covolume on 𝐷𝐶. Theorem 4.2. Let 𝐶0 be an arithmetic chain over a maximal order O in a definite quaternion algebra over Q . Let 𝐺 be a finite index subgroup of PU𝑞(O) . Then there exists a constant 𝜅 > 0such that, as 𝜖 > 0tends to 0, the number 𝜓𝐶0,𝐺 (𝜖) of chains modulo 𝐺∞in the 𝐺-orbit of 𝐶0with 𝑑Cyg-diameter at least 𝜖is equal to 35 223 36𝐷2 𝐴Covol𝐺(𝐶0)[PU𝑞(O)∞:𝐺∞] 𝜋6𝑚𝐶0,𝐺 𝑚𝐴|O×|2Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]𝜖−10 1+O(𝜖𝜅), where 𝑚𝐶0,𝐺 is the order of the pointwise stabiliser of 𝐷𝐶0in 𝐺. Recall that the center cen(𝐶) of a finite chain 𝐶 is the image of ∞=[ 1 : 0 : 0 ] under the reflexion on 𝐿𝐶 . The following result is an equidistribution result in the quaternionic Heisenberg group of the centers of the arithmetic chains in a given orbit under (a finite index subgroup of) PU𝑞(O). 62 Rigidity, counting and equidistribution Theorem 4.3. Let 𝐶0,𝐺and 𝑚𝐶0,𝐺 be as in Theorem 4.2. As 𝜖 > 0tends to 0, we have 𝑚𝐶0,𝐺𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺] 35 224 36Covol𝐺(𝐶0)𝜖10 ∑︁ 𝐶∈𝐺·𝐶0 diam𝑑Cyg (𝐶)≥𝜖 Δcen(𝐶) ∗ ⇀HaarHeis7. As in Theorem 4.1, there exist 𝜅 > 0and ℓ∈N such that for every smooth function 𝜓 with compact support on Heis7 , there is an error term in this equidistribution result when the measures on both sides are evaluated on 𝜓 , of the form O(𝑠−𝜅k𝜓kℓ) where k𝜓kℓ is the Sobolev norm of 𝜓. We begin by a technical result used in the proofs of the above theorems, which does not require the assumption 𝑛= 2. Recall that 𝑑00 Cyg is the modified Cygan distance defined in Section 2. Lemma 4.4. For every 𝑚-chain 𝐶in H𝑛 H, we have diam𝑑Cyg (𝐶)=√2 diam𝑑00 Cyg (𝐶). Proof. If 𝐶 is a vertical 𝑚 -chain, then both diameters are +∞ . We hence assume that 𝐶 is finite. Since the Heisenberg translations and rotations preserve 𝑑Cyg and 𝑑00 Cyg , and by the transitivity properties of the Heisenberg translations and rotations on the set of 𝑚 -chains (see Section 3.2), we may assume that 𝐶 is a Euclidean sphere centered at ( 0 , 0 ) with dimension 4 𝑚− 1, contained in the horizontal plane H𝑛−1×{ 0 } of Heis4𝑛−1 . Since the Heisenberg dilations (𝜁, 𝑢) ↦→ (𝜆𝜁, 𝜆2𝑢) with 𝜆 > 0are homotheties of ratio 𝜆 for 𝑑Cyg and 𝑑00 Cyg, we may assume that the radius of 𝐶is equal to 1. For every (𝜁, 0 ) ∈ 𝐶 , we thus have 𝑑Cyg((𝜁, 0 ),( 0 , 0 )) = 1by Equation (2.7) , hence diam𝑑Cyg (𝐶) ≤ 2by the triangle inequality. Since 𝑑Cyg((𝜁, 0),(−𝜁, 0)) =𝑑Cyg((𝜁, 0)·(𝜁, 0),(0,0)) =𝑑Cyg((2𝜁, 0),(0,0)) =2, we have diam𝑑Cyg (𝐶)=2. Using the transitivity properties of Sp(𝑛− 1 ) on the unit sphere 𝐶 of the Euclidean space H𝑛−1 in the same way as in the proof of [ 20 , Lem. 8] in the complex hyperbolic case, we may assume that 𝑛=3, and that diam𝑑00 Cyg (𝐶)=sup 𝑢∈H,𝜙∈[0, 𝜋 ]:n(𝑢)=1 𝑑00 Cyg (1,0,0),(𝑢cos 𝜙, sin 𝜙, 0). 63 J. Parkkonen & F. Paulin By a computation similar to the one in [ 20 , Lem. 8], using Equation (2.8) and the fact that 4n(Im 𝑢)=4− (tr 𝑢)2for any unit quaternion 𝑢, we have 𝑑00 Cyg (1,0,0),(𝑢cos 𝜙, sin 𝜙, 0)2 =𝑑00 Cyg (0,0,0),(−1,0,0)·(𝑢cos 𝜙, sin 𝜙, 0)2 =𝑑00 Cyg (0,0,0),𝑢cos 𝜙−1,sin 𝜙, −2 cos 𝜙Im 𝑢)2 =(2−cos 𝜙tr 𝑢)2+4 cos2𝜙n(Im 𝑢) ((2−cos 𝜙tr 𝑢)2+4 cos2𝜙n(Im 𝑢))1 2+ (2−cos 𝜙tr 𝑢) =2 1 (1+cos2𝜙−cos 𝜙tr 𝑢)1 2+2−tr 𝑢cos 𝜙 2(1+cos2𝜙−cos 𝜙tr 𝑢) . As 1+cos2𝜙−cos 𝜙tr 𝑢≤2−cos 𝜙tr 𝑢≤4, we have 𝑑00 Cyg (1,0,0),(𝑢cos 𝜙, sin 𝜙, 0)2≤2. Furthermore, the equality holds when 𝑢=1and 𝜙=𝜋. This proves the result.  Proof of Theorem 4.2 and Theorem 4.3. The diameter of a chain for the Cygan distance is invariant under the stabiliser in PU𝑞 of the horosphere 𝜕H1 , hence is invariant under 𝐺∞. The counting function 𝜓𝐶0,𝐺 is thus well defined. Note that H1 is a horoball centered at the fixed point of a parabolic element in PU𝑞(O) (take the vertical Heisenberg translation by ( 0 , 2 𝑢) for any nonzero 𝑢∈O∩Im H ). We will apply Theorem 4.1 with Γ = 𝐺 , with 𝐷−=H1 , which is hence a horoball centered at the fixed point of a parabolic element in 𝐺 , and with 𝐷+=𝐷𝐶0 , which is the quaternionic geodesic line in H2 Hwith boundary at infinity equal to 𝐶0. In particular 𝑚+=𝑚𝐶0,𝐺. Let us compute the constant 𝑐(𝐷−, 𝐷+) appearing in the statement of Theorem 4.1. We have Vol(𝐺\H2 H)=[PU𝑞(O):𝐺]Vol(PU𝑞(O)\H2 H), where, by [22, Thm. 1·4], Vol(PU𝑞(O)\H2 H)=𝜋4𝑚𝐴 175 213 35Ö 𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1), and by [22, Lem. 8·4], Vol(Γ𝐷−\𝐷−)=[PU𝑞(O)∞:𝐺∞]Vol(PU𝑞(O)H1\H1)=𝐷2 𝐴[PU𝑞(O)∞:𝐺∞] 160 |O×|2.(4.2) By definition, we have Vol(Γ𝐷+\𝐷+)=16 Covol𝐺(𝐶0), 64 Rigidity, counting and equidistribution since the sectional curvature of 𝐷+ is constant − 4and 𝐷+ has real dimension 4. We hence have 𝑐(𝐷−, 𝐷+)=35 213 36𝐷2 𝐴Covol𝐺(𝐶0)[PU𝑞(O)∞:𝐺∞] 𝜋6𝑚𝐶0,𝐺𝑚𝐴|O×|2Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺].(4.3) Let 𝑔∈𝐺 be such that the quaternionic geodesic line 𝑔𝐷+ is disjoint from H1 (which is the case except for 𝑔 in finitely many double classes in 𝐺H1\𝐺/𝐺𝐷+ ). Let 𝛿𝑔 be the common perpendicular from H1 to 𝑔𝐷+ . Its length ℓ(𝛿𝑔) is the minimum of the distances from H1 to a geodesic line between two points of 𝜕∞(𝑔𝐷+)=𝑔𝐶0 . Hence, by Lemmas 2.1 and 4.4, we have ℓ(𝛿𝑔)=min 𝑥,𝑦 ∈𝑔𝐶0, 𝑥≠𝑦𝑑(H1,]𝑥, 𝑦[) =−max 𝑥,𝑦 ∈𝑔𝐶0, 𝑥≠𝑦ln 𝑑00 Cyg(𝑥, 𝑦) √2 =−ln diam𝑑00 Cyg (𝑔𝐶0) √2=−ln diam𝑑Cyg (𝑔𝐶0) 2.(4.4) Respectively by the definition of the counting function 𝜓𝐶0,𝐺 in the statement of Theorem 4.2, since the stabiliser of 𝐶0 in 𝐺 is equal to 𝐺𝐷𝐶0=𝐺𝐷+ , by Equation (4.4) , by Theorem 4.1, and by Equation (4.3), we have, as 𝜖 > 0tends to 0, 𝜓𝐶0,𝐺 (𝜖) =card 𝐺∞\{𝐶∈𝐺·𝐶0: diam𝑑Cyg (𝐶) ≥ 𝜖} =card{[𝑔] ∈ 𝐺∞\𝐺/𝐺𝐷𝐶0: diam𝑑Cyg (𝑔𝐶0) ≥ 𝜖} =card n[𝑔] ∈ 𝐺H1\𝐺/𝐺𝐷𝐶0:ℓ(𝛿𝑔) ≤ −ln 𝜖 2o+O(1) =N𝐷−,𝐷+−ln 𝜖 2+O(1)=𝑐(𝐷−, 𝐷+)𝑒−10 ln 𝜖 21+O(𝑒𝜅ln 𝜖 2) =35 223 36𝐷2 𝐴Covol𝐺(𝐶0)[PU𝑞(O)∞:𝐺∞] 𝜋6𝑚𝐶0,𝐺𝑚𝐴|O×|2Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]𝜖−10 1+O(𝜖𝜅). This proves Theorem 4.2. Let us now prove Theorem 4.3. We apply the equidistribution result in Equation (4.1) of the origins or(𝛿𝑔) of the common perpendiculars 𝛿𝑔 from 𝐷−=H1 to the images 𝑔𝐷+ for 𝑔∈𝐺 . As 𝑠→ +∞ , we hence have, using Equations (4.3) and (4.2), 𝑚𝐶0,𝐺𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺] 35 217 36Covol𝐺(𝐶0)𝑒−10𝑠 ∑︁ [𝑔]∈𝐺/𝐺𝐷+:ℓ(𝛿𝑔)≤𝑠 Δor(𝛿𝑔)∗ ⇀vol𝜕H1.(4.5) 65 J. Parkkonen & F. Paulin Let 𝑓 : 𝜕∞ H 2 H− {∞} =Heis7→𝜕H1 be the orthogonal projection map, which is the homeomorphism (𝑤0, 𝑤) ↦→ (𝑤0+1 2, 𝑤) . The pushforward of the Haar measure HaarHeis7by 𝑓is 𝑓∗HaarHeis7=8 vol𝜕H1,(4.6) see for example the end of the proof of Theorem 8·3 in [22]. Note that, for every chain 𝐶 , if 𝑟𝐶 is the reflexion on the quaternionic projective line containing 𝐶 , then the geodesic line from ∞ to cen(𝐶)=𝑟𝐶(∞) , being invariant under 𝑟𝐶 , is orthogonal to the quaternionic geodesic line with boundary at infinity 𝐶 . Hence for every 𝑔∈𝐺, we have 𝑓−1(or(𝛿𝑔)) =cen(𝑔𝐶0). Let us use in Equation (4.5) the change of variables 𝑠=−ln 𝜖 2 and the continuity of the pushforward of measures by 𝑓−1 . By Equations (4.4) and (4.6) , as 𝜖 > 0tends to 0, we obtain that the measures 𝑚𝐶0,𝐺𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺] 35 224 36Covol𝐺(𝐶0)𝜖10 ∑︁ [𝑔]∈𝐺/𝐺𝐷+ diam𝑑Cyg (𝑔𝐶0)≥𝜖 Δcen(𝑔𝐶0) weak-star converge to the Haar measure HaarHeis7. This proves Theorem 4.3.  Example 4.5. Let 𝐶0=[𝑤0 : 0 : 1 ] ∈ P2 r(H) : tr 𝑤0= 0  be the standard vertical chain in 𝜕∞ H 2 H , which is the intersection of 𝜕∞ H 2 H with the quaternionic projective line 𝐿𝐶0=[𝑧0:𝑧1:𝑧2] ∈ P2 r(H):𝑧1=0. An element ±©« 𝑎 𝛾∗𝑏 𝛼 𝑀 𝛽 𝑐 𝛿∗𝑑ª®®¬ of PU𝑞 preserving the quaternionic geodesic line 𝐿𝐶0∩ H 2 H satisfies 𝛼𝑤0+𝛽= 0for all 𝑤0∈H with tr 𝑤0> 0. Thus, 𝛼=𝛽= 0, and Equations (2.5) (or rather the similar equations obtained by the formula 𝑋𝑋∗=𝐼𝑛+1 instead of 𝑋∗𝑋=𝐼𝑛+1 ) imply that 𝛾=𝛿= 0. Using again Equations (2.5) , we see that the stabiliser of 𝐿𝐶0 consists of the elements ©« 𝑎0𝑏 0𝑀0 𝑐0𝑑ª®®¬ such that tr(𝑐𝑎)=tr(𝑑𝑏)=0,𝑐𝑏 +𝑎𝑑 =1and 𝑀∈O×. Thus, CovolPU𝑞(O)(𝐶0)=𝜋2 1080 Ö 𝑝|𝐷𝐴(𝑝−1)(𝑝2+1) 66 Rigidity, counting and equidistribution by [5, Thm. 2.5]. The pointwise stabiliser of 𝐶0 in PU𝑞(O) consists of the diagonal elements with 𝑎=𝑑=±1and 𝑀∈O×, giving 𝑚𝐶0,PU𝑞(O)=|O×|. Theorems 4.2 and 4.3 then give 𝜓𝐶0,PU𝑞(O)(𝜖)=189 220𝐷2 𝐴 𝜋4𝑚𝐴|O×|3Î𝑝|𝐷𝐴(𝑝3−1)𝜖−10 1+O(𝜖𝜅), and 𝜋4𝑚𝐴|O×|Î𝑝|𝐷𝐴(𝑝3−1) 189 221 𝜖10 ∑︁ 𝐶∈PU𝑞(O)·𝐶0: diam𝑑Cyg 𝐶≥𝜖 Δcen(𝐶)∗ ⇀HaarHeis7. References [1] Yves Benoist and Hee Oh. Effective equidistribution of 𝑆 -integral points on symmetric varieties. Ann. Inst. Fourier, 62(5):1889–1942, 2012. [2] Yves Benoist and Jean-François Quint. Stationary measures and invariant subsets of homogeneous spaces II. J. Am. Math. Soc., 26(3):659–734, 2013. [3] Arthur L. Besse. Einstein manifolds. Classics in Mathematics. Springer, 2008. [4] Olivier Biquard. Quaternionic contact structures. In Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics (Roma, 1999), pages 23–30. World Scientific, 2001. [5] Stefan Breulmann and Volker Helmke. The covolume of quaternion groups on the four-dimensional hyperbolic space. Acta Arith., 77(1):9–21, 1996. [6] Anne Broise-Alamichel, Jouni Parkkonen, and Frédéric Paulin. Equidistribution and counting under equilibrium states in negative curvature and trees. Applications to non-Archimedean Diophantine approximation, volume 329 of Progress in Mathematics. Birkhäuser, 2019. [7] Élie Cartan. Sur le groupe de la géométrie hypersphérique. Comment. Math. Helv., 4:158–171, 1932. [8] William Duke. Rational points on the sphere. Ramanujan J., 7(1-3):235–239, 2003. [9] Jordan S. Ellenberg, Philippe Michel, and Akshay Venkatesh. Linnik’s ergodic method and the distribution of integer points on spheres. In Automorphic representations and L-functions, volume 22 of Tata Institute of Fundamental Research Studies in Mathematics, pages 119–185. Tata Institute of Fundamental Research, 2013. 67