ANoteonLocalRigidity.PDFVIP

Geometriae Dedicata 107: 111–131, 2 4. 111 # 2 4 Kluwer Academic Publishers. Printed in the Netherlands. A Note on Local Rigidity N. BERGERON1 and T. GELANDER2 1 ´ ´ ˆ Unite Mixte de Recherche 8628 du CNRS, Laboratoire de Mathematiques, Bat. 425, ´ Universite Paris-Sud, 91405 Orsay Cedex, France. e-mail: Nicolas.Bergeron@math.u-psud.fr 2Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Heaven, CT 06520, U.S.A. e-mail: tsachik.gelander@yale.edu (Received: 8 March 2 4) Abstract. The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way. Key words. lattices, local rigidity, semisimple Lie groups. 1. Introduction Let G be a semisimple Lie group, and let G 4G be an irreducible lattice. We denote by RðG; GÞ the space of deformations of G in G, i.e. the space of all homo- morphisms G ! G with the topology of pointwise convergence. Recall that G is called locally rigid if there is a neighborhood O of the inclusion map r : G ! G in RðG; GÞ such that any r 2 O is conjugate to r . In other words, this means that if G is generated by the finite set S, then there is an identity neighborhood U G such that if r: G ! G is a homomorphism such that rðgÞ 2 g U for any g 2 S, then there is some g 2 G for which rðgÞ ¼ ggg1 for any g 2 G (such r is called a trivial deformation). Local rigidity was first proved by Selberg [17] for uniform lattices in the case G ¼ SLn ðRÞ; n 53, and by Calabi [3] for uniform lattices in the case G ¼ POðn; 1Þ ¼ IsomðHnÞ; n 53. Then, Weil [22] generalized these results to any uniform