C. Maclachlan

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.

Arithmeticity, discreteness and volume

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of P SL(2, C). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple.

The arithmetic structure of tetrahedral groups of hyperbolic isometries

Introduction . Polyhedra in 3-dimensional hyperbolic space which give rise to discrete groups generated by reflections in their faces have been investigated in [14], [17], [29] and in the case of tetrahedra there are precisely nine compact non-congruent ones with dihedral angles integral submultiples of π [14]. These polyhedral groups give rise to hyperbolic 3-orbifolds and examples of these have been studied, for example, in [3], [15], [18], [24], [25].

Generalised Fibonacci manifolds

Fibonacci manifolds have a hyperbolic structure which may be defined via Fibonacci numbers. Using related sequences of Lucas numbers, other 3-manifolds are constructed, their geometric structures determined, and a curious relationship between the homology and the invariant trace-field examined.

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.

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.

Compact hyperbolic 4-manifolds of small volume

We prove the existence of a compact non-orientable hyperbolic 4-manifold of volume 32π 2 /3 and a compact orientable hyperbolic 4-manifold of volume 64π 2 /3, obtainable from torsion-free subgroups of small index in the Coxeter group [5,3,3,3]. At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.

Fuchsian Subgroups of Bianchi Groups

A maximal non-elementary Fuchsian subgroup of a Bianchi group PSL(2, Od) has an invariant circle or straight line under its linear fractional action on the complex plane, to which is associated a positive integer D, the discriminant, which, in turn, is an invariant of the wide commensurability class of the Fuchsian subgroup. In this paper, for all Bianchi groups, we classify the conjugacy classes of these maximal Fuchsian subgroups by determining the number with given discriminant.

Bounds for the Number of Automorphisms of a Compact Non-Orientable Surface

The paper shows that for every positive integer

Torsion in Arithmetic Fuchsian Groups

p > 2