Déformations de réseaux dans certains groupes résolubles

Abstract

We aim to study local rigidity and deformations for the following class of groups: the semidirect product Γ = Zⁿ $\ times$_A Z where n ≥ 2 is an integer and A is a hyperbolic matrix in SL(n,Z), considered first as a lattice in the solvable Lie group G = Rⁿ $\ times$_A R, then as a subgroup of the semisimple Lie group SL(n+1,R). We will notably show that, although Γ is locally rigid neither in G nor in H, it is locally SL(n+1,R)-rigid in G in the sense that every small enough deformation of Γ in G is conjugated to Γ by an element of SL(n+1,R).