Alan W. Reid

The Bianchi groups are separable on geometrically finite subgroups

Let d be a square free positive integer and Od the ring of integers in Q( p id). The main result of this paper is that the groups PSL(2;Od) are

Essential closed surfaces in bounded 3-manifolds

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S 9<M of a closed, orientable connected surface S is essential if it is injective at the level of fundamental groups and the group i*rr1 (S) cannot be conjugated into a subgroup 7rr(coM) of -rr(M), where &oM is a component of OM. This latter condition is equivalent to the statement that the image of the surface S cannot be freely homotoped into OM. One of the main results of this paper is the following:

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.

Geodesics and commensurability classes of arithmetic hyperbolic

This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas recalled in §2.1 and §2.2, each element of QL(M) ∪ QL(M ′) is a rational multiple of the logarithm of a real algebraic number. As noted by Prasad and Rapinchuk in [9], the Gelfond Schneider Theorem [1] implies that a ratio of such logarithms is transcendental if it is irrational. Thus if ` ∈ QL(M)−QL(M ′) then `/`′ is transcendental for all non-zero `′ ∈ QL(M ′). Recently Prasad and Rapinchuk have shown in [9] that if M is an arithmetic hyperbolic manifold of even dimension, then QL(M) and the commensurability class of M determine one another. In addition, they have shown that this is not always true for arithmetic hyperbolic 5-manifolds. However, they have announced a proof that for all locally symmetric spaces associated to a specified absolutely simple Lie group, there are only finitely many commensurability classes of arithmetic lattices giving rise to a given rational length spectrum. It is known (see [4] pp. 415–417) that for closed hyperbolic manifolds, the spectrum of the Laplace-Beltrami operator action on L2(M), counting multiplicities, determines the set of lengths of closed geodesics on M (without counting multiplicities). Hence Theorem 1.1 implies:

3

We provide, for hyperbolic and flat 3{manifolds, obstructions to bounding hyperbolic 4{manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.

-manifolds

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.

On the geometric boundaries of hyperbolic 4-manifolds.

We show that hyperbolic 3-manifolds have residually simple fundamental group.

Arithmeticity, discreteness and volume

We give a complete classification of the unknotting tunnels in 2-bridge link complements, proving that only the upper and lower tunnels are unknotting tunnels. Moreover, we show that the only strongly parabolic tunnels in 2-cusped hyperbolic 3-manifolds are exactly the upper and lower tunnels in 2-bridge knot and link complements. From this, it follows that the upper and lower tunnels in 2-bridge knot and link complements must be isotopic to geodesics of length at most ln(4), where length is measured relative to maximal cusps. Moreover, the four dual unknotting tunnels in a 2-bridge knot complement, which together with the upper and lower tunnels form the set of all known unknotting tunnels for these knots, must each be homotopic to a geodesic of length at most 6ln(2).

SIMPLE QUOTIENTS OF HYPERBOLIC 3-MANIFOLD GROUPS

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].

Unknotting tunnels in two-bridge knot and link complements

We show that all closed flat n-manifolds are dieomorphic to a cusp crosssection in a nite volume hyperbolic n + 1-orbifold. AMS Classication 57M50; 57R99