Algebraic $K$-theory of hyperbolic manifolds

F. T. Farrell, L. E. Jones
1986 Bulletin of the American Mathematical Society  
Let r = TT\M where M is a complete hyperbolic manifold with finite volume. We announce (among other results) that Wh T = 0 where Wh T is the Whitehead group of T. We also announce WI12 r = 0, k 0 (ZT) = 0, Kn (ZT) = 0 (for n > 0), and Wh n T 0 Q = 0 (for all n). We calculate the weak homotopy type of the stable topological concordance space C(M), and hence Waldhausen's Wh PL -theory (cf. [22]) of M, in terms of simpler stable concordance spaces. When M is compact, the calculation is in terms of
more » ... tion is in terms of ^(S 1 ) where S 1 is the circle. A connected complete Riemannian manifold M is called weakly admissible if there exist positive real numbers a < b such that all the sectional curvatures of M are less than -a and bigger than -6. A weakly admissible manifold is admissible if it has finite volume. In particular, all complete locally symmetric spaces having finite volume and strictly negative sectional curvatures are admissible Riemannian manifolds. These are precisely the real, complex, quaternionic and Cayley complete hyperbolic manifolds of finite volume. All complete manifolds of constant negative sectional curvature and finite volume occur among these; in fact, they are the complete real hyperbolic manifolds of finite volume. The purpose of this paper is to announce the calculation of the algebraic K-theory of admissible manifolds. We start by stating that the Whitehead group Wh TT\M of the fundamental group of an admissible manifold M vanishes. Actually, we proceed to formulate and state a bit more general result. A group T is K-flat if Wh(r©C n ) = 0 for all nonnegative integers n where C n denotes the free abelian group of rank n. The Bass-Heller-Swan formula [3] implies WhT = 0, K 0 {ZT) = 0 and it_ n (Zr) = 0 provided T is If-flat and n > 0. A smooth fiber bundle F -• E ~y M is admissible if
doi:10.1090/s0273-0979-1986-15412-1 fatcat:idcocepqizccfl2f7jibjwrq2y