D. D. Long

The Burau representation is not faithful for n ≥ 6

This should be contrasted with the method of [6] which in this context involves a passage from n to n + 2. In fact our methods produce examples which are substantially simpler than those constructed in [6] at least measured by crossing number. This obstruction can also be used to give elements in the kernel of the representation obtained from the Burau reprcscntation of the five strand braid group by reducing all coefficients modulo 2. Some extra ideas arc required in order to keep some of the linear algebra under control and we need to appeal to the following result regarding Burau matrices:

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:

REMARKS ON THE A-POLYNOMIAL OF A KNOT

This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into . The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have non-trivial polynomial. We include several open questions.

Computing Varieties of Representations of Hyperbolic 3-Manifolds into SL(4, ℝ)

The geometric structure on a closed orientable hyperbolic 3- manifold determines a discrete faithful representation ρ of its fundamental group into SO+(3, 1), unique up to conjugacy. Although Mostow rigidity prohibits us from deforming ρ, we can try to deform the composition of ρ with inclusion of SO+(3, 1) into a larger group. In this sense, we have found by exact computation a small number of closed manifolds in the Hodgson- Weeks census for which ρ deforms into SL(4,ℝ), thus showing that the hyperbolic structure can be deformed in these cases to a real projective structure. In this paper we describe the method for computing these deformations, particular attention being given to the manifold Vol3.

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

It is shown that with nitely many exceptions, the fundamental group obtained by Dehn surgery on a one cusped hyperbolic 3{manifold contains the fundamental group of a closed surface.

On the geometric boundaries of hyperbolic 4-manifolds.

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

Some surface subgroups survive surgery

We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.