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 $\Gamma=\bm{Z}^{n}\rtimes_{A}\bm{Z}$  where $n\geq 2$  is an integer and $A$  is a hyperbolic matrix in $\SL{n}{Z}$, considered first as a lattice in the solvable Lie group $G=\bm{R}^{n}\rtimes_{A}\bm{R}$, then as a subgroup of the semisimple Lie group $\SL{n+1}{R}$. We will notably show that, although $\Gamma$  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 $\Gamma$  in $G$  is conjugated to $\Gamma$  by an element of $\SL{n+1}{R}$.