$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume
$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume
复制标题
$ L^2 $-有限体积双曲流形的扭力
DOI：
--
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
T. Schick
中科院分区：
文献类型：
--
作者：
W. Lück;T. Schick
抽象的。假设
$ \bar{M} $ 是一个有边界的紧连通奇维流形，其内部 M 具有有限体积的完全双曲度量。我们将证明
$ L^2 $-拓扑扭转
$ \bar{M} $ 和
$ L^2 $-黎曼流形 M 的解析挠率是相等的。特别是，
$ L^2 $-拓扑扭转
$ \bar{M} $ 与 M 的双曲体积成正比，比例常数仅取决于维度，并且已知在奇数维度 [HS] 中不为零。在维度 3 中，这证明了猜想 [Lü2，猜想 2.3] 或 [LLü，猜想 7.7]，它给出了完整的计算
$ L^2 $-紧致拓扑扭转
$ L^2 $-非循环 3-流形，允许几何 JSJT 分解。¶在附录中，我们给出了将 Cheeger-Müller 定理扩展到具有边界的流形的反例：如果度量不是边界附近的乘积，则一般来说，即使边界的欧拉特征消失，解析挠率和拓扑挠率也不相等。
Abstract. Suppose
$ \bar{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
$ L^2 $-topological torsion of
$ \bar{M} $ and the
$ L^2 $-analytic torsion of the Riemannian manifold M are equal. In particular, the
$ L^2 $-topological torsion of
$ \bar{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
$ L^2 $-topological torsion of compact
$ L^2 $-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.