Daryl Cooper

Automorphisms of free groups have finitely generated fixed point sets

An automorphism of a finitely generated free group F extends to a homeomorphism of the end completion E of F. The set of fixed points of this homeomorphism is finitely generated in a certain sense. In particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense. The emphasis on E instead of F in this paper is analogous to Thurston’s study of surface groups where measured laminations are studied instead of simple closed curves. The techniques in this paper arose from a study of the behaviour of an automorphism under iteration, that is, the dynamics of the automorphism; see [ 11. I thank Bill Thurston for explaining some of his ideas to me for analysing automorphisms of free groups. In particular the result on bounded cancellation below is due to Thurston and Matt Grayson. Peter Scott asked whether the fixed subgroup is always finitely generated in [3], and Gersten first answered this question in [2] using quite different methods.

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.

Some surface subgroups survive surgery

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.

Representation theory and the A-polynomial of a knot

Abstract This paper surveys various aspects of the two variable polynomial of knots defined by consideration of the representations of the fundamental group of the complement into SL(2,ℂ).

Free actions of finite groups on rational homology 3-spheres

We show that any finite group can act freely on a rational homology 3-sphere.

Bundles and finite foliations.

By a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume We recall that if is a Kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space Otherwise is geometrically in nite If happens to be a surface group then we say it is quasi Fuchsian if the limit set for the group action is a Jordan curve C and preserves the components of S n C The starting point for this work is the following theorem which is a combination of theorems due to Marden Thurston and Bonahon

Deforming convex projective manifolds

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for non-compact (G,X)-manifolds of the openess of their holonomies.

The marked length spectrum of a projective manifold or orbifold

A strictly convex real projective orbifold is equipped with a natural Finsler metric called a Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to projective duality. A corollary is the existence of non-isometric diffeomorphic strictly convex projective manifolds (and orbifolds) that are isospectral. This corollary follows from work of Goldman and Choi, and Benoist.