L2-Betti numbers of mapping tori and groups

Abstract

IN his preprint [12, p. 152 and p. 1561 Gromov states the following two conjectures: Let a compact aspherical manifold M be fibered over the circle S’. Then all LZ-Betti numbers b,(M) are trivial. Let { 1) + A -+ F + rc + { 1) be an extension of infinite groups which are fundamental groups of finite aspherical CW-complexes. Then the first L2-Betti number b,(F) is trivial. We will give affirmative answers to these conjectures. The first conjecture follows from Theorem 2.1 which states that all L2-Betti numbers b,(T,) of a mapping torus T, of an endomorphism f of a finite CIV-complex F vanish. We prove in Theorem 4.1 for an extension {l} -+ A -+ F + x + { 1) of finitely presented groups that the first L2-Betti number b,(F) vanishes provided that A is infinite and n contains Z as a subgroup. This implies the second conjecture above. Let F be an infinite finitely presented group with trivial first L2-Betti number br (F). As applications we show in Theorem 5.1 that a closed 4-manifold with I as fundamental group satisfies X(M) 2 lo(iU)I for X(M) the Euler characteristic and o(M) the signature. This generalizes a result of Johnson and Kotschick [16]. We prove in Theorem 6.1 that the deficiency of F satisfies def(F) I 1. L2-Betti numbers were introduced by Atiyah [l]. In Section 1 we recall their definitions and basic properties from the topological point of view. They also have an analytic meaning, namely, the p-th L2-Betti number of a closed Riemannian manifold measures the size of the space of harmonic L2-integrable smooth p-forms of the universal covering [7]. For general information and applications of L2-Betti numbers, and in particular of conditions that determine when they vanish, the reader may refer for example to [l], [4], (IS], [6], C71, PI, Clll, C121, C181, C191, WI and WI. The paper is organized as follows: