Rigidity, counting and equidistribution of quaternionic Cartan chains

Parkkonen, Jouni; Paulin, Frédéric

Annales mathématiques Blaise Pascal, Tome 28 (2021) no. 1, pp. 45-69.

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.

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.

Classification : 11E39, 11F06, 11N45, 20G20, 53C17, 53C55
Keywords: counting, equidistribution, Cartan chain, quaternionic Heisenberg group, Cygan distance, sub-Riemannian geometry, quaternionic hyperbolic geometry
Mot clés : comptage, équidistribution, chaîne de Cartan, groupe de Heisenberg quaternionien, distance de Cygan, géométrie sous-riemanienne, géométrie hyperbolique quaternionienne

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