Changing topological type of compression bodies through cone manifolds

Abstract

Homeomorphism types of compression bodies form the vertices of a graph where two vertices are joined by an edge if one compression body is obtained by gluing a $2$-handle onto the other. Motivated by earlier work of Lackenby and Purcell on geodesicity of unknotting tunnels for hyperbolic links, we show that it is possible to realise all of the edges in the graph of compression bodies by paths of cone manifold holonomy groups such that the handle that is glued in is obtained as a limit of singular arcs of cone angle increasing from $0$ to $ 2\pi $. We apply standard techniques from the theory of $ \mathrm{CAT}(0) $ spaces, and do not rely on the harmonic deformation theory of Hodgson and Kerckhoff. Along the way we prove a generalisation of a classic theorem of Koebe and Maskit on existence of function groups which implies existence results for reflex angled hyperbolic cone structures on a wide range of compression bodies.