Counting problems for special-orthogonal Anosov representations  [ Problèmes de comptage pour les représentations d’Anosov dans le groupe spécial orthogonal ]
Annales de l'Institut Fourier, à paraître, 59 p.

Pour des entiers positifs p et q soit G:=PSO(p,q) le groupe projectif spécial-orthogonal indéfini de signature (p,q). Nous étudions des problèmes de comptage dans l’espace symétrique Riemannien X G de G et dans l’espace hyperbolique pseudo-Riemannien p,q-1 . Soit SX G une copie totalement géodésique de X PSO(p,q-1) . Nous examinons l’orbite de S sous l’action d’un sous-groupe de G de type projectivement Anosov. Pour certains choix d’une telle copie géodésique, nous montrons que le nombre de points dans cette orbite qui se trouvent à une distance maximale t de S est fini et asymptotiquement purement exponentiel lorsque t tend vers l’infini. Nous fournissons une interprétation de ce résultat dans p,q-1 , comme l’asymptotique de la quantité de segments géodésiques de type espace de longueur maximale t dans l’orbite d’un point.

For positive integers p and q let G:=PSO(p,q) be the projective indefinite special-orthogonal group of signature (p,q). We study counting problems in the Riemannian symmetric space X G of G and in the pseudo-Riemannian hyperbolic space p,q-1 . Let SX G be a totally geodesic copy of X PSO(p,q-1) . We look at the orbit of S under the action of a projective Anosov subgroup of G. For certain choices of such a geodesic copy we show that the number of points in this orbit which are at distance at most t from S is finite and asymptotic to a purely exponential function as t goes to infinity. We provide an interpretation of this result in p,q-1 , as the asymptotics of the amount of space-like geodesic segments of maximum length t in the orbit of a point.

Reçu le : 2018-12-05
Révisé le : 2019-09-02
Accepté le : 2019-09-18
Classification : 22E40,  53C35
Mots clés: Représentations d’Anosov, comptage, espaces symétriques
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     author = {Carvajales, Le\'on},
     title = {Counting problems for special-orthogonal Anosov representations},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Carvajales, León. Counting problems for special-orthogonal Anosov representations. Annales de l'Institut Fourier, à paraître, 59 p.

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