L 2 - Torsion of Hyperbolic Manifolds of Finite Volume

Suppose M is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L2-topological torsion of M and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M is proportional to the hyperbolic volume of M , with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, Conjecture 7.7] which gives a complete calculation of the L2-topological torsion of compact L2-acyclic 3-manifolds which admit a geometric JSJT-decomposition. In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. 0 Introduction In this paper we study L2-analytic torsion of a compact connected manifold M with boundary such that the interior M comes with a complete hyperbolic metric of finite volume. Notation 0.1. Let m be the dimension of M . From hyperbolic geometry we know [BP, Chapter D3] that M can be written as M = M0 ∪∂M0 E0 , where M0 is a compact manifold with boundary and E0 is a finite disjoint union of hyperbolic ends [0,∞)×Fj . Here each Fj is a closed flat manifold and the metric on the end is the warped product du2 + e−2udx2 with dx2 the metric of Fj . Of course, we can make the ends smaller and also write M = MR ∪∂MR ER R ≥ 0 , Vol. 9, 1999 L2-TORSION OF HYPERBOLIC MANIFOLDS OF FINITE VOLUME 519 where ER is the subset of E0 consisting of the components [R,∞)×Fj . We define TR = MR+1 ∩ER R ≥ 0 . Denote by M̃R, ẼR, . . . the inverse images of MR, ER, . . . under the universal covering map M̃ →M . Correspondingly, for the universal covering we have M̃ = M̃R ∪∂M̃R ẼR . For ẼR we get and fix coordinates such that each component is [R,∞)×Rm−1 with warped product metric du2 + e−2udx2 , where dx2 is the Euclidean metric on Rn. Each MR is a compact connected Riemannian manifold with boundary which is L2-acyclic (see Corollary 6.5). Its absolute L2-analytic torsion T (2) an (MR) is defined. The manifold M has no boundary and finite volume and its L2-analytic torsion is defined although it is not compact (Remark 1.9). We will recall the notion of L2-analytic torsion in section 1. The main result of this paper is Theorem 0.2. Let M be a compact connected manifold with boundary such that the interior M comes with a complete hyperbolic metric of finite volume. Suppose that the dimension m of M is odd. Then we get, if R tends to infinity lim R→∞ T (2) an (MR) = T (2) an (M) . Remark 0.3. For compact Riemannian manifolds with boundary and with a product metric near the boundary, analytic torsion and topological torsion differ by (ln 2)/2 times the Euler characteristic of the boundary. This is a result of Lück in the classical situation [Lü1] and of Burghelea et al. [BuFK] for the L2-version. In particular, analytic torsion does not depend on the metric, as long as it is a product near the boundary and the manifold is acyclic or L2-acyclic, respectively. In Appendix A we give examples which show that this is not longer the case for arbitrary metrics. This answers an old question of Cheeger’s [C, p. 281]. Of course it also implies that the above extension of the CheegerMüller theorem to manifolds with boundary is not true in general. This requires additional care in the chopping and exhausting process described above. We will explain the strategy of proof for 0.2 in section 1. Next we discuss consequences of Theorem 0.2 and put it into context with known results. 520 W. LÜCK AND T. SCHICK GAFA We will explain in section 1 that the comparison theorem for L2-analytic and -topological torsion for manifolds with and without boundary of Burghelea, Friedlander, Kappeler and McDonald [BuFK], [BuFKM] now implies Theorem 0.4. Let M be a compact connected manifold with boundary such that the interior M comes with a complete hyperbolic metric of finite volume. Then T (2) top(M) = T (2) an (M) . The computations of Lott [L, Proposition 16] and Mathai [M, Corollary 6.7] for closed hyperbolic manifolds extend directly to hyperbolic manifolds without boundary and with finite volume since Hm is homogeneous. (Notice that we use for the analytic torsion the convention in Lott which is twice the logarithm of the one of Mathai.) Hence Theorem 0.4 implies Theorem 0.5. Let M be a compact connected manifold with boundary such that the interior M comes with a complete hyperbolic metric of finite volume. Then there is a dimension constant Cm such that T (2) an (M) = Cm · vol(M) . Moreover, Cm is zero, if m is even, and C3=−1/3π and (−1)C2m+1>0. Remark 0.6. The last statement is a result of Hess [HS] and answers the question of Lott [L] whether C2m+1 is always nonzero. For closed manifolds, the proportionality follows directly from the computations of Lott and Mathai and the comparison theorem for L2-analytical and topological torsion of Burghelea et.al. [BuFKM] (as is also observed there). Now Theorem 0.4 and Theorem 0.5 together with [Lü2, Theorem 2.1] imply Theorem 0.7. Let N be a compact connected orientable irreducible 3-manifold with infinite fundamental group possessing a geometric JSJTdecomposition such that the boundary of N is empty or a disjoint union of incompressible tori. Then all L2-Betti numbers of N vanish and all Novikov-Shubin invariants are positive. Moreover, we get T (2) top(N) = −1 3π · r ∑