Rigidity, counting and equidistribution of quaternionic Cartan chains

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.


Introduction
The sphere at infinity ∞ of a negatively curved symmetric space 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 endow the sphere at infinity of with arithmetic structures, and problems of equidistribution of rational points or subvarieties in ∞ , 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 , 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 . 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 , and, analogously to the complex hyperbolic case treated in [20], an effective equidistribution result for the arithmetic 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 ↦ → its conjugation, n : ↦ → its reduced norm, tr : ↦ → + its reduced trace. Let be the quaternionic Hermitian form on the right vector space H 3 over H defined by ( 0 , 1 , 2 ) = − tr( 0 2 ) + n( 1 ), and PU its projective unitary group. It is the isometry group of the quaternionic hyperbolic plane H 2 H , realised as the negative cone of in the right projective plane P 2 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 in P 2 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.  We endow the metabelian simply connected real Lie group Heis 7 with its Cygan distance Cyg , which is the unique left-invariant distance such that Cyg (( 0 , ), (0, 0)) = (4 n( 0 )) 1 4 . The chains contained in Heis 7 are ellipsoids, and have a natural center cen( ) and radius (see Section 3).
Let be a definite ( ⊗ Q R = H) quaternion algebra over Q, with discriminant . Let O be a maximal order in . We refer for instance to  An arithmetic chain 0 bounds in H 2 H a homothetic copy of the real hyperbolic space of dimension 4. We denote by Covol( 0 ) the volume of the quotient of this real hyperbolic space, normalised to have sectional curvature −1, by the stabiliser PU (O) 0 of 0 in PU (O), and by 0 the order of the pointwise stabiliser of this real hyperbolic space in PU (O). We endow the real Lie group Heis 7 with its Haar measure Haar Heis 7 normalised in such a way that the total mass of the induced measure on the quotient of Heis 7 by its We refer to Section 4 for a version with congruences and error terms, and a more developped study of explicit examples of arithmetic chains.

Its boundary at infinity is
For every ∈ N, let be the identity × matrix. Let . The Cygan distance Cyg on Heis 4 −1 is the unique left-invariant distance on the real Lie group Heis 4 −1 such that or equivalently Cyg (( 0 , ), (0, 0)) = (4 n( 0 )) 1 4 by Equation (2.6). We introduce (see [19,20] in the complex case) the modified Cygan distance Cyg , as the unique left-invariant map from Though not actually a distance, the map Cyg is symmetric and satisfies For every nonempty bounded subset of Heis 4 −1 , we define the diameter of for this almost distance as Note that the Cygan distance and the modified Cygan distance are invariant under Heisenberg translations and rotations, and that for every > 0, the Heisenberg dilation ℎ is a homothety of ratio for both distances.
Proof. By the invariance under Heisenberg translations of H 1 , of the distance in H H and of the modified Cygan distance, we may assume that , the geodesic line from ( 0 , ) to (0, 0) is, up to translation at the source, the map 0 , : The point 0 , ( ) belongs to the horosphere H ( ) , where, since tr 0 = n( ), . Let = n( 0 ) 1/2 be the norm of the vector 0 and the angle between the vectors 1 and 0 in the Euclidean space H. Then the map ↦ → ( ) = 2 2 2 4 2 + 2 2 cos + 1 reaches its maximum at 2 = 1 . Since tr 0 = n( ), the value of this maximum is The result then follows from Equation (2.4).

Chains
In this section, we define the quaternionic Cartan chains and give their elementary geometric properties, see also [24]. In the complex case, the notion of chain is attributed to von Staudt by [7]. The exposition follows the one of [10] in the complex case. We fix ∈ {1, . . . , − 1}. When = is a finite hyperchain, that is, when ≠ 0, then is a codimension 4 ellipsoid in the Euclidean space H −1 × Im H, whose vertical projection is the Euclidean sphere in H −1 with real codimension 1 and equation n( )−tr( · ( −1 ))+tr( 0 −1 ) = 0 in the unknown , with center −1 and radius

A vocabulary of chains
This radius of the Euclidean sphere Π ( ) is called the radius of the finite hyperchain . The map Π | from to Π ( ) is a homeomorphism. When = 0 and 0 Similarly, a finite chain is a 3-dimensional ellipsoid in the Euclidean space H −1 ×Im H, whose vertical projection is a Euclidean 3-sphere in H −1 . In particular, any chain is homeomorphic to the 3-sphere S 3 .

Transitivity properties of PU on chains
Through any two distinct projective points belonging to ∞ H H passes one and only one quaternionic projective line, and this projective line meets H H . Hence through two distinct points of ∞ H H passes one and only one chain. By Witt's theorem, the group PU acts transitively on the set of quaternionic projective spaces of dimension meeting H H , hence it acts transitively on the set of -chains.
Note that two -chains having the same vertical projection differ by a vertical Heisenberg translation, that the group generated by Heisenberg translations and rotations acts transitively on the set of vertical -chains, and that PB (that contains the Heisenberg dilations, rotations and translations) acts transitively on the set of finite -chains.
The next result gives the topological structure of a family of chains, called a fan in the complex hyperbolic case (see for instance [10, p. 131] Proof. By the transitivity properties of PU , we may assume that = ∞. Hence is a finite chain, and by the transitivity properties of the Heisenberg translations, we may assume that is a Euclidean sphere of dimension 4 − 1 contained in the horizontal space

Reflexions on chains
The chains are fixed point sets at infinity of natural isometries of H H , that we now describe. If is a proper quaternionic projective subspace of P r (H) meeting H H , there exists a unique involution in PU with fixed point set , called the reflexion on . Note that the set of fixed points of (1) The reflexions and commute.

Description of the center and radius of chains
We now define and study the centers of chains, whose equidistribution we will prove in Section 4.

A von Staudt-Cartan rigidity theorem
The following theorem shows that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. This is a quaternionic version of the result of Cartan in the complex case (see for instance [10,Thm. 4.3.12]), close to von Staudt's fundamental theorem of real projective geometry. A similar proof shows that an injective map from ∞ H H to itself, such that any three points belong to a same chain if and only if their images by belong to a same chain, is the restriction of an element of PU .

Relation with the hyper CR structure
In this subsection, we give a characterisation of the chains in terms of the natural hyper CR structure on ∞ H H . We refer for example to [3] and [14] for background on hyperkähler manifolds and hyper CR manifolds, respectively.
We endow the manifold P r (H) with its natural hyperkähler structure, and we denote by In the following result, we denote by = 1 + 2 + 3 the standard coordinate in Im H, and by d the tautological (Im H)-valued 1-form on Im H, so that for every ∈ Im H, the map d :   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 = 2 and that is a Euclidean sphere with center (0, 0) and radius 1 in the horizontal subspace {( , ) ∈ H −1 × H : = 0}. Since the Im H-valued 1-form | is invariant under the Heisenberg rotations, the volume form 1 ∧ 2 ∧ 3 on is invariant under the Heisenberg rotations. Since the only measure on invariant under the Heisenberg rotations is, up to a scalar multiple, the Lebesgue measure on the Euclidean sphere , the measure is a multiple of the Lebesgue measure on . This can also be proved by a direct computation: On the Euclidean sphere , with = 0 + 1 + 2 + 3 , we have Since the barycenter of this measure is exactly the origin (0, 0), which is the center of the finite chain , this proves Assertion (1).
(2). First assume that is the standard vertical chain (3). First assume that is the Euclidean 3-sphere ( , ) ∈ H −1 × Im H : n( 1 ) = 2 and = 2 = · · · = −1 = 0 , and that = ( = (− , 0, . . . , 0), = 0). Note that is the radius of the finite chain . By the properties of the exponential map of the Lie group of unit quaternions, whose tangent space at the identity element 1 is Im H, the smooth map The uniqueness of up to postcomposition by a Heisenberg rotation preserving and , and the extension to the other chains, follow as previously from the fact that the chains are transverse to the quaternionic contact structure on Heis 4 −1 and by invariance of the calibration under the Heisenberg translations and rotations.

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]. where Γ acts diagonally on Γ × Γ. When Γ has no torsion, N − , + ( ) is the number (with multiplicities coming from the fact that Γ ± \ ± is not assumed to be embedded in Γ\H H ) of the common perpendiculars of length at most between the images of − and + in Γ\H H . The following statement is a special case of [22,Thm. 8·1]. We denote by Δ the unit Dirac mass at a point . .
There exists > 0 such that, as → +∞, Furthermore, the origins of the common perpendiculars from − to the images of + under the elements of Γ equidistribute in − to the induced Riemannian measure: As → +∞, we have   As in Theorem 4.1, there exist > 0 and ℓ ∈ N such that for every smooth function with compact support on Heis 7 , there is an error term in this equidistribution result when the measures on both sides are evaluated on , of the form O( − ℓ ) where ℓ is the Sobolev norm of .
We begin by a technical result used in the proofs of the above theorems, which does not require the assumption = 2. Recall that Cyg is the modified Cygan distance defined in Section 2.

Lemma 4.4. For every -chain in H
Proof. If is a vertical -chain, then both diameters are +∞. We hence assume that is finite. Since the Heisenberg translations and rotations preserve Cyg and Cyg , and by the transitivity properties of the Heisenberg translations and rotations on the set of -chains (see Section 3.2), we may assume that is a Euclidean sphere centered at (0, 0) with dimension 4 − 1, contained in the horizontal plane H −1 × {0} of Heis 4 −1 . Since the Heisenberg dilations ( , ) ↦ → ( , 2 ) with > 0 are homotheties of ratio for Cyg and Cyg , we may assume that the radius of is equal to 1. For every ( , 0) ∈ , we thus have Cyg (( , 0), (0, 0)) = 1 by Equation (2.7), hence diam Cyg ( ) ≤ 2 by the triangle inequality. Since Using the transitivity properties of Sp( − 1) on the unit sphere of the Euclidean space H −1 in the same way as in the proof of [20,Lem. 8] in the complex hyperbolic case, we may assume that = 3, and that Note that H 1 is a horoball centered at the fixed point of a parabolic element in PU (O) (take the vertical Heisenberg translation by (0, 2 ) for any nonzero ∈ O ∩ Im H ). We will apply Theorem 4.1 with Γ = , with − = H 1 , which is hence a horoball centered at the fixed point of a parabolic element in , and with + = 0 , which is the quaternionic geodesic line in H 2 H with boundary at infinity equal to 0 . In particular + = 0 , . Let us compute the constant ( − , + ) appearing in the statement of Theorem 4.1. We have Vol( Let ∈ be such that the quaternionic geodesic line + is disjoint from H 1 (which is the case except for in finitely many double classes in H 1 \ / + ). Let be the common perpendicular from H 1 to + . Its length ℓ (  [