Complex curves and surgery

Complex Sprays and Complex Curves

After defining what is meant by a complex spray X on a complex manifold M , we introduce the notion of a spray complex curve associated to X. Several equivalent formulations are derived and we give necessary and sufficient conditions for M to admit spray complex curves for X through each point and in each direction. Refinements of this result are then derived for some special cases, for example...

Elliptic Curves and Complex Multiplication

Kleinian groups and the complex of curves

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have ‘bounded geometry’ (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed ...

Lipschitz Geometry of Complex Curves

We describe the Lipschitz geometry of complex curves. For the most part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, b...

Exceptional Surgery Curves in Triangulated 3-manifolds

For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope σ is ‘exceptional’ if the resulting 3-manifold MK(σ) is reducible or a solid torus, or the core of the surgery solid torus has finite order in π1(MK(σ)). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between σ and the meridian slope is more than one, and a homolog...