no. 3
Counting problems for special-orthogonal Anosov representations
[Problèmes de comptage pour les représentations d’Anosov dans le groupe spécial orthogonal]
Carvajales, León1, 2
1 Sorbonne Université - Campus Pierre et Marie Curie Institut de Mathématiques de Jussieu 4, place Jussieu - Boite Courrier 247 75252 Paris Cedex 05 Paris (France)
2 Universidad de la República - Facultad de Ciencias Centro de Matemática Iguá 4225, 11400 Montevideo (Uruguay)
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1199-1257.

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 ℍ p,q-1 . Soit S⊂X 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 ℍ 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 ℍ p,q-1 . Let S⊂X 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 ℍ 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 :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3333
Classification : 22E40, 53C35
Keywords: Anosov representations, counting, symmetric spaces
Mots-clés : Représentations d’Anosov, comptage, espaces symétriques

Carvajales, León 1, 2

1 Sorbonne Université - Campus Pierre et Marie Curie Institut de Mathématiques de Jussieu 4, place Jussieu - Boite Courrier 247 75252 Paris Cedex 05 Paris (France)
2 Universidad de la República - Facultad de Ciencias Centro de Matemática Iguá 4225, 11400 Montevideo (Uruguay)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Carvajales, León. Counting problems for special-orthogonal Anosov representations. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1199-1257. doi : 10.5802/aif.3333. https://aif.centre-mersenne.org/articles/10.5802/aif.3333/

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