### Déformations de réseaux dans certains groupes résolubles

Cédric ROUSSEAU
Source: J. Math. Soc. Japan Volume 60, Number 2 (2008), 397-421.

#### 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 $\SL{n+1}{R}$.

First Page:
Primary Subjects: 22E25, 22E40
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1212156656
Digital Object Identifier: doi:10.2969/jmsj/06020397
Zentralblatt MATH identifier: 1144.22009

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