On the spectrum of a large subgroup of a semisimple group

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January 2008, 2(1): 15-42. doi: 10.3934/jmd.2008.2.15

On the spectrum of a large subgroup of a semisimple group

1.	IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France

Received September 2006 Revised September 2007 Published October 2007

Abstract
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We consider a semi-simple algebraic group $\mathbf G$ defined over a local field of zero characteristic and we denote by $G$ the group of its $k$-rational points. For $\Gamma$ a "large" sub-semigroup 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 $\mathbb R$ or $\mathbb Q_{p}$, allows to use the Lie algebras techniques. The properties of the limit set of $\Gamma$ are developed ; they play an important role in the proofs.

Keywords: orbit closure., Zariski-dense, proximal, randomwalk, boundary.

Mathematics Subject Classification: Primary: 37H15; Secondary: 22E30,3.

Citation: Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15

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