Journal of the Mathematical Society of Japan

Déformations de réseaux dans certains groupes résolubles


Full-text: Open access


We aim to study local rigidity and deformations for the following class of groups: the semidirect product Γ= Z n A Z   where n2   is an integer and A   is a hyperbolic matrix in SL( n,Z ) , 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( 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 ) .

Article information

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

First available in Project Euclid: 30 May 2008

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 22E25: Nilpotent and solvable Lie groups 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

local rigidity lattices in solvable Lie groups group cohomology


ROUSSEAU, Cédric. Déformations de réseaux dans certains groupes résolubles. J. Math. Soc. Japan 60 (2008), no. 2, 397--421. doi:10.2969/jmsj/06020397.

Export citation


  • E. Calabi, On compact, Riemannian manifolds with constant curvature I, Proc. Sympos. Pure Math., III, Amer. Math. Soc., Providence, R. I., 1961, pp.,155–180.
  • E. Calabi and E. Vesentini, On compact, locally symmetric Kälher manifolds, Ann. of Math. (2), 71 (1960), 472–507.
  • D. Fisher, Local rigidity of group actions: past, present, future, preprint, 9–10.
  • A. Guichardet, Cohomologie des groupes topologiques et des algèbres de Lie, CEDIC/Fernand Nathan, Collection “Textes mathématiques”, 1980.
  • M. S. Raghunathan, On the first cohomology of discrete subgroups of semisimple Lie groups, Amer. Jour. Math., 41 (1965), 103–139.
  • M. S. Raghunathan, Discrete subgroups of Lie groups, Springer, New-York, 1972.
  • A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, pp.,147–164.
  • A. Weil, On discrete subgroups of Lie groups, Ann. of Math., 72 (1960), 369–384.
  • A. Weil, On discrete subgroups of Lie groups II, Ann. of Math., 75 (1962), 578–602.
  • A. Weil, Remarks on the cohomology of groups, Ann. of Math., 80 (1964), 149–157.