Dehn function

Abstract: The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

Abstract: In the late 1970’s, Thurston dramatically changed the nature of 3-manifold theory with the introduction of his Geometrisation Conjecture, and by proving it in the case of Haken 3-manifolds [23]. The conjecture for general closed orientable 3-manifolds remains perhaps the most important unsolved problem in the subject. A weaker form of the conjecture [19] deals with the fundamental group of a closed orientable 3-manifold. It proposes that either it contains Z ⊕ Z as a subgroup or it is word hyperbolic, in the sense of Gromov [11]. Word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality. They are of fundamental importance in geometric group theory and have very many useful properties.

Abstract: If F is a finitely generated discrete group and G a complex algebraic Lie group, the G-character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in G. Marc Culler and Peter Shalen developed the theory of SL2(C)-character varieties of finitely generated groups and applied their results to study the topology of 3-dimensional manifolds in the papers [6], [7], [8]. Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. The Mostow rigidity theorem implies that the holonomy representation p: iri(M) Isom+(H3) _ PSL2(C) is unique up to conjugation and taking complex conjugates. The orientability of M can be used to show 1 lifts to a representation p E Hom(-ri (M), SL2(C)) whose character determines an essentially unique point of Xp of X(iri(M)), the SL2(C)-character variety of irl(M). Culler and Shalen [8] proved that the component X0 of X(X1ri(M)) which contains X' is a curve. One of their major contributions was to show how X0 determines a norm on H1(WM; R) which encodes many topological properties of M. In particular it provides information on the Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL2(C)-character variety of a connected, compact, orientable, irreducible 3manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a seminorm. The first half of this paper is devoted to the development of the general theory of -Culler-Shalen seminorms defined for curves of PSL2(C)-characters. By working over PSL2(C) we obtain a theory that is more generally applicable than its SL2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimen-