Portal de Periódicos da CAPES

On the First Betti Number of a Constant Negatively Curved Manifold

1976; Princeton University; Volume: 104; Issue: 2 Linguagem: Inglês

10.2307/1971046

1939-8980

John J. Millson,

Geometry and complex manifolds

constant negatively curved n-dimensional manifolds with arbitrarily large first Betti number. In fact we show that any constant negatively curved manifold whose fundamental group is an arithmetic group commensurable with the group of units of a quadratic form admits a finite covering with first Betti number not equal to zero; in particular, the examples given by Borel at the end of [4] all admit such coverings. Previous to this a few examples were known in low dimensions (cf. Vinberg [14]) with examples up to dimension 5 in the compact case. However, his construction uses hyperbolic Coxeter groups which exist only in low dimensions. Rather than attack the problem algebraically by computing the abelianized fundamental group by group theory an approach that appears hopeless except for the above low dimensional examples, we take a geometric approach and construct explicit nonbounding codimension 1 cycles and then appeal to Poincare duality. Although these examples are of obvious geometric interest their main significance is group theoretic. The vanishing theorem of Kajdan [6] has as a consequence that the first Betti number of all compact locally irreducible, locally symmetric spaces of rank greater than 2 vanishes. This was extended by S. P. Wang [15] and Kostant [8] who show that Kajdan's criterion applies to all compact locally irreducible locally symmetric spaces except those associated with SO(n, 1) and SU(n, 1). The results of this paper show that the vanishing theorem does not hold for SO(n, 1). Whether or not it holds for SU(n, 1) remains unsolved. Our results are of interest in connection with the congruence subgroup problem. Bass, Milnor, Serre [1] show that if an arithmetic group F satisfies the congruence subgroup property that every subgroup of finite index contains a congruence subgroup, then: