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$$L^ 2$$-cohomology of geometrically infinite hyperbolic 3-manifolds. (English) Zbl 0873.57014
Let $$M$$ be a connected hyperbolic 3-manifold which is topologically tame, i.e. it is diffeomorphic to the interior of a compact 3-manifold with boundary. The ends of such a manifold can be characterized as cusps, flares and tubes. If there are no tubes, it is called geometrically finite and otherwise geometrically infinite. The paper deals with the question for a topologically tame hyperbolic 3-manifold whether there are non-zero square-integrable harmonic 1-forms and whether zero lies in the spectrum of the Laplacian acting on $$\Lambda^1(M)/\ker(d)$$. If $$M$$ is compact, the questions are equivalent and the answer is positive if and only if $$H^p(M;\mathbb{C})$$ is non-zero. The situation in the non-compact case is much more complicated and involved, the questions are not equivalent and the answer depends on more than the topology.
We summarize some of the results of the paper: In the case that $$M$$ is the mapping torus of a pseudo-Anosov diffeomorphism $$f$$ of a surface $$S$$, zero lies in the spectrum of the Laplacian acting on $$\Lambda^1(M)/\ker(d)$$ if and only if $$f^*:H^1(S,\mathbb{R})\to H^1(S,\mathbb{R})$$ has an eigenvalue of norm one. If $$M$$ has zero injectivity radius, then the essential spectrum of the Laplacian acting on $$\Lambda^1(M)/\ker(d)$$ is $$[0,\infty)$$.
Let $$M$$ be a topologically tame hyperbolic 3-manifold with positive injectivity radius such that the ends of $$M$$ are incompressible. Given a tubular end there is a model for it of the shape $$[0,\infty)\times S$$ for an appropriate surface $$S$$. Denote by $$\Gamma(H^1)$$ the Hilbert space of measurable maps $$f:[0,\infty)\to H^1(S,\mathbb{R})$$ such that $$\int^\infty_0\langle f(t),f(t)\rangle_t dt<\infty$$ holds for a certain family of inner products $$\langle\;,\;\rangle_t$$ on $$H^1(S,\mathbb{R})$$. Let $$\Gamma'(H^1)$$ be the subspace of $$\Gamma(H^1)$$ of those functions which are absolutely continuous and satisfy $$\partial_tf\in\Gamma(H^1)$$. Then zero lies in the spectrum of the Laplacian acting on $$\Lambda^1(M)/\ker(d)$$ if and only if each end of $$M$$ is tubular and the corresponding operator $$\partial_t:\Gamma'(H^1)\to \Gamma(H^1)$$ is surjective. Further sufficient criterions are proven. The kernel of $$\Delta_1$$ for such $$M$$ is finite-dimensional.
Reviewer: W.Lück (Münster)

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds
##### Keywords:
hyperbolic 3-manifold; ends; topologically tame; Laplacian
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