On the spectrum of a large subgroup of a semisimple group

Yves Guivarc'H (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We consider a semisimple algebraic group G defined over a local field of zero characteristic and we denote by G the group of its k-rational points. For Gamma a "large" subsemigroup of G we define a closed subgroup < Spec Gamma > associated with Gamma, and we show that < Spec Gamma > is large in a certain sense. This allows us to study the Gamma-orbit closures for certain Gamma-actions. The analytic structure of closed subgroups of G, over R or Q(p), allows us to use the Lie algebras techniques. The properties of the limit set of Gamma are developed; they play an important role in the proofs.

Mots clés: Zariski-dense; boundary; proximal; random walk; orbit closure

Domaines: Théorie des groupes [math.GR]

Yves Guivarc'H. On the spectrum of a large subgroup of a semisimple group. Journal of modern dynamics, 2008, 2 (1), pp.15-42. ⟨10.3934/jmd.2008.2.15⟩. ⟨hal-00360714⟩