Arithmetic Hyperbolic 3-manifolds Research Articles

Statistical mechanics of 2+1 gravity from Riemann zeta function and Alexander polynomial: exact results

In the recent publication [J. Geom. Phys. 33 (2000) 23], we have demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work, we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of two-dimensional liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston’s terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just the ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self-homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemann surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic, i.e. the discrete Lie groups of motion are arithmetic.

The finite subgroups of maximal arithmetic kleinian groups

Nous calculons, en fonction des parametres arithmetiques decrits par Borel, les sous-groupes finis d'un groupe de Klein arithmetique maximal. Ceci est notamment appliquable a l'etude des 3-varietes arithmetiques hyperboliques.

2-generator arithmetic Kleinian groups

Abstract We show that among the infinitely many conjugacy classes of finite co-volume Kleinian groups generated by two elements of finite order, there are only finitely many which are arithmetic. In particular there are only finitely many arithmetic generalized triangle groups. This latter result generalizes a theorem of Takeuchi.

Sur l'homologie de cycles géodésiques dans des variétés hyperboliques compactes

The notion of an l-geodesic cycle in a compact hyperbolic n-manifold M generalises, in dimension l, the one of a closed geodesic. In this Note we show that when l ≥n/2, such a cycle lifts to a finite cover of M as an embedded totally geodesic submanifold non zero homologous. It enables us to prove that the compact hyperbolic manifolds constructed by Gromov and Piateski-Shapiro (see [4]) have infinite virtual Betti numbers and to give a new proof of the same fact for the compact arithmetic hyperbolic manifolds constructed by Borel in [3].

Two-Generator Arithmetic Kleinian Groups II

Extending earlier work, we establish the finiteness of the number of two-generator arithmetic Kleinian groups with one generator parabolic and the other either parabolic or elliptic. We also identify all the arithmetic Kleinian groups generated by two parabolic elements. Surprisingly, there are exactly 4 of these, up to conjugacy, and they are all torsion free. 1991 Mathematics Subject Classification 30F40, 20H10.

Geodesic intersections in arithmetic hyperbolic 3-manifolds

It was shown by Chinburg and Reid that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert showed that almost all quasi-Fuchsian 3-manifolds have all closed geodesics simple and disjoint. The natural conjecture arose that the Chinburg-Reid examples also had disjoint geodesics. Here we show that this conjecture is both almost true (they have no geodesics that intersect except at right angles) and spectacularly false (any pair of closed geodesics admits infinitely many closed geodesics which intersects both geodesics of the pair perpendicularly). The latter statement is shown to be true for all closed arithmetic hyperbolic 3-manifolds. Section 0 Introduction By a hyperbolic n-manifold we shall mean a complete orientable n-dimensional Riemannian manifold all of whose sectional curvatures are −1. If M is a hyperbolic 3-manifold, the universal cover of M can be identified with H, the upper half-space model of hyperbolic 3-space, andM is realized asH/Γ for some Γ a discrete torsionfree subgroup of Isom(H3). Now Isom(H3) can be identified with PSL(2,C) ( which in turn is isomorphic to PGL(2,C)), and Γ is called a Kleinian group. In the sequel we will only be interested in the case when M is closed, in which case Γ is referred to as cocompact. Of interest to us is the structure of the set of closed geodesics in closed hyperbolic 3-manifolds. The motivation comes from the following. A closed geodesic in a (closed) hyperbolic n-manifold is simple if it has no self-intersections, and nonsimple otherwise. In dimension 2 every closed hyperbolic manifold has a non-simple closed geodesic. However in dimension 3 the situation is much more complex. Many closed hyperbolic 3-manifolds contain immersions of totally geodesic surfaces and so there are non-simple closed geodesics. In [JR] examples were given of closed hyperbolic 3-manifolds containing a non-simple closed geodesic but having no immersed totally geodesic surface. However it was shown in [CR] that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert [BW] showed that almost all 3-manifolds arising as the quotient of H 1991 Mathematics Subject Classification. Primary: 57M50, Secondary: 30F40.

The behaviour of eigenstates of arithmetic hyperbolic manifolds

In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.

A Characterization of Arithmetic Subgroups ofSL (2, ℝ) andSL (2, ℂ)

A Kleinian group is a discrete subgroup of PSL(2,C). As such it acts on 3-dimensional hyperbolic space H3. A Kleinian group G is said to have finite covolume if H3/G has finite volume. An interesting subclass of Kleinian groups of finite covolume is that of arithmetic Kleinian groups. Such a group is constructed as follows. Let k be a finite extension of Q with exactly one pair of complex conjugate embeddings (it may have many real embeddings), A a quaternion algebra over k which is ramified at all the real embeddings of k and O an order of A. We denote by O1 the elements of O of reduced norm 1. Via the complex embedding we have a map ρ:A↪M(2,C) and, in particular, O1↪SL(2,C). It can be shown that ρ(O1) is a discrete subgroup of SL(2,C), and projectivizing gives a discrete subgroup of PSL(2,C) which has finite covolume. An arithmetic Kleinian group is a subgroup of PSL(2,C) commensurable with some group Pρ(O1). These groups have many interesting properties and form a good set of examples with which to test conjectures in the theory of Kleinian groups and hyperbolic 3-manifolds. With this in mind it is an interesting problem to classify such groups in elementary terms. This was done by the reviewer in his thesis [see also C. Maclachlan and A. W. Reid, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 251–257;], following the ideas of K. Takeuchi for characterizing arithmetic subgroups of PSL(2,R) [see J. Math. Soc. Japan 27 (1975), no. 4, 600–612;]. The main result of the paper under review is to rework these characterizations. As an application, the authors detail which orbifolds arising as cyclic branch covers of the Whitehead link are arithmetic.

Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups

Arithmetic Fuchsian and Kleinian groups can all be obtained from quaternion algebras (see [2,12]). In a series of papers ([8,9,10,11]), Takeuchi investigated and characterized arithmetic Fuchsian groups among all Fuchsian groups of finite covolume, in terms of the traces of the elements in the group. His methods are readily adaptable to Kleinian groups, and we obtain a similar characterization of arithmetic Kleinian groups in §3. Commensurability classes of Kleinian groups of finite co-volume are discussed in [2] and it is shown there that the arithmetic groups can be characterized as those having dense commensurability subgroup. Here the wide commensurability classes of arithmetic Kleinian groups are shown to be approximately in one-to-one correspondence with the isomorphism classes of the corresponding quaternion algebras (Theorem 2) and it easily follows that there are infinitely many wide commensurability classes of cocompact Kleinian groups, and hence of compact hyperbolic 3-manifolds.

Embedding arithmetic hyperbolic manifolds

We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $\mathrm{mod}~2$ Abelian cover can.