Anosov maps, Polycyclic groups and Homology

Abstract

Topology Vol. IO pp. 177-183. Pergamon Press, 1971. Printed in Cc-at Britain. ANOSOV MAPS, POLYCYCLIC GROUPS AND HOMOLOGY MORRIS W. HIRSCH~ (Receioed 15 Septenlber 1970) INTRODUCTION LET M DENOTE a compact Riemannian manifold. A C’ map f: A4 + M is called Anosoo if there exists a continuous splitting TM = E”@ E” of the tangent bundle of M, and constants C > 0, 1. > 1, such that Tf(E”) = E”, TJ(E’) c ES, and furthermore for all positive integers m and tangent vectors X E TM: This condition is independent ITf”(X) I 2 CJ. IXl if XEEU. of the Riemannian metric. Anosov maps have been studied in [I, 2, 7, 1 I], and elsewhere. Examples of Anosov maps can be obtained from a Lie group G, a discrete subgroup I with G/T compact, and an endomorphism 4: G ---t G such that #(I) c I; see [8]. If the derivative of q5 at the identity of G has no eigenvalues of absolute value one, then the map G/I -+ G/lY induced by 4 is an Anosov map. All known Anosov maps are intimately related to maps of this type. As an example take G = R” and r = Z”, the integer lattice; then $I is defined by an n x n integer matrix A and G/T is the n-torus T”. In this case we can identify A with the linear transformation f*: induced by f on the first homology H,(T”; R) -+ H,(T”; R) group with real coefficients. John Franks [2] has proved that iff: T” -+ T” is any Anosov has no root of unity among its eigenvalues. diffeomorphism, then I;, Theorem 1 extends Franks’ result to Anosov maps on a wider class of manifolds which includes all nilmanifolds. As an application. many manifolds that do not admit Anosov diffeomorphisms are constructed. For example: the Cartesian product of the Klein bottle and a torus. 7 Supported in part by NSF Grant GP 22723.