The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.
If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = {pi}_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F with the same number of boundary components as F .
We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative i
nteger weights, 14 for each 3-simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the homology map is the unit ball, B, of the Thurston norm. Applications of this approach include (1) an algorithm to compute B and hence the Thurston norm of any homology class, (2) an explicit exponential bound on the number of vertices of B in terms of the number of simplices in the triangulation, (3) an algorithm to determine the fibred faces of B and hence an algorithm to decide whether a 3-manifold fibres over the circle.
