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 $\Gamma =Z^n{⋊}_{A}Z$ where $n\ge 2$ is an integer and $A$ is a hyperbolic matrix in $SL\left(n,Z\right)$, considered first as a lattice in the solvable Lie group $G=R^n{⋊}_{A}R$, then as a subgroup of the semisimple Lie group $SL\left(n+1,R\right)$. We will notably show that, although $\Gamma$ is locally rigid neither in $G$ nor in $H$, it is locally $SL\left(n+1,R\right)$-rigid in $G$ in the sense that every small enough deformation of $\Gamma$ in $G$ is conjugated to $\Gamma$ by an element of $SL\left(n+1,R\right)$.