H-loxodromic subgroups
SOUS-GROUPES H-LOXODROMIQUES
Abstract
Consider k a finite extension of Q_p, with p a prime number. Let H be a finite index subgroup of k^* and G be the group SL(n,k) with its Zariski topology of Q _p-group. We investigate the existence of a subgroup of G which is Zariski-dense and such that each of its elements has a spectrum included in H. A necessary and sufficient condition is obtained: such a subgroup exists if and only if either -1 belongs to H or the dimension n is not congruent to 2 modulo 4.
On considère une extension finie k de Qp, avec p un nombre premier, H un sous-groupe d'indice fini de k * et le groupe SL(n, k). Nous montrons que SL(n, k) admet un sous-groupe Qp-Zariski-dense dont toutes les matrices ont leur spectre inclus dans H si et seulement si soit −1 est dans le sous-groupe H, soit n n'est pas congru à 2 modulo 4. Abstract (H-loxodromic subgroups). — Consider k a finite extension of Qp, with p a prime number. Let H be a finite index subgroup of k * and G be the group SL(n, k) with its Zariski topology of Qp-group. We investigate the existence of a subgroup of G which is Zariski-dense and such that each of its elements has a spectrum included in H. A necessary and sufficient condition is obtained: such a subgroup exists if and only if either −1 belongs to H or the dimension n is not congruent to 2 modulo 4.
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