Arithmetic of Hyperbolic Manifolds

By a hyperbolic 3-manifold we mean a complete orientable hyperbolic 3-manifold of finite volume, that is a quotient H/Γ with Γ ⊂ PSL2C a discrete subgroup of finite covolume (here briefly “a Kleinian group”). Among hyperbolic 3-manifolds, the arithmetic ones form an interesting, and in many ways more tractable, subclass. The tractability comes from the availability of arithmetic tools and invariants. For example, an arithmetic manifold M = H/Γ is determined up to commensurability by its defining field k (a number field with exactly one complex place) and quaternion algebra A (which is ramified at all real places of k ). Any such pair (k,A) determines a unique commensurability class of arithmetic hyperbolic 3-manifolds. One aim of this paper is to try to extend arithmetic considerations to more general hyperbolic 3-manifolds. For example, a commensurability invariant pair (k(M), A(M)) consisting of a non-totally-real number field and a quaternion algebra over it is defined for any M = H/Γ (Sects. 2 and 3; we also write (k(Γ), A(Γ)) ), but it fails to be a complete commensurability invariant of M —non-commensurable M can have the same A(M) (see Sect. 10 for examples). Nevertheless, k(M) and A(M) do contain useful information—for instance (Theorem 3.2 and Proposition 3.3) ramification of A(M) at a finite prime forces subgroups of Γ to have non-trivial abelianizations, A(M) gives a good amphicheirality test (Proposition 3.4), and k(Γ) composes under amalgamation of Kleinian groups along a non-elementary Kleinian group (Theorem 2.8 and [NR1]) and is therefore a mutation invariant. The trace field Q(tr Γ) of Γ is not a commensurability invariant and the field k(Γ) is in fact the smallest field among the trace fields of finite index subgroups of Γ. We call it the invariant trace field of Γ. We show that the trace field Q(tr Γ) is a Galois (Z/2)-extension of the invariant trace field k(Γ) and there is a precise Galois relationship between subgroups of Γ and their trace fields, the largest subgroup of Γ with trace field k(Γ) being normal with quotient (Z/2) (Theorem 2.2). It turns out that the existence of parabolic elements in Γ facilitates many arithmetic questions. For example, in this case A(Γ) is just the matrix algebra M2(k(Γ)). Moreover, k(Γ) equals the field generated by the tetrahedral parameters of the ideal tetrahedra of any ideal triangulation of H/Γ (Theorem 2.4). If one conjugates Γ so that three parabolic fixed points are at 0, 1, and ∞ in C ∪ {∞} = ∂H , then k(Γ) is also the field generated by all parabolic fixed points (Lemma 2.5). As another example, an arithmetic orbifold with cusps cannot have geodesics shorter than 0.431277313 (cf. Theorem 4.6 and Corollary 4.7); a corresponding result for com-