On the Non-vanishing of the First Betti Number of Hyperbolic Three Manifolds
نویسنده
چکیده
We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1, D), where D is a quaternion division algebras defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture of Thurston. The proof uses the characterisation of the image of solvable base change by the author, and the construction of cusp forms with non-zero cusp cohomology by Labesse and Schwermer.
منابع مشابه
The Growth Rate of the First Betti Number in Abelian Covers of 3-manifolds
We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds M with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of M , for there to exist a family of Zn covering spaces, Mn, in which β1(Mn) increases linearly with n....
متن کاملThe Growth Rate of the First Betti Number in Abelian Covers of 3-manifolds Tim D. Cochran and Joseph Masters
We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds M with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of M , for there to exist a family of Zn covering spaces, Mn, in which β1(Mn) increases linearly with n....
متن کاملCovering Spaces of Arithmetic 3-orbifolds
This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3-orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3-manifolds; namely whether a closed hyperbolic 3-manifold is virtually Haken. Various strengthenings of this have also been widely studied. Of specific to interest to us is the question of whe...
متن کاملSome Examples of Aspherical Symplectic Four-manifolds
We give examples of aspherical symplectic 4-manifolds, which are formal in the sense of rational homotopy theory but have degenerate Lefschetz pairings. (preliminary version) Compact Kähler manifolds have a number of topological properties which are not shared by general symplectic manifolds, see [1] for example. In particular, Kähler manifolds have the following cohomological properties: 1. th...
متن کامل4 F eb 1 99 9 Weyl structures with positive Ricci tensor
We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity condition. Thus we obtain a generalization of the vanishing theorem of Bochner, which has a particularly simple form in dimension 4. As a corollary we obtain that if the canonical Weyl structure on a compact Hermitian surface is non-exact, the symmetri...
متن کامل