Changing topological type of compression bodies through cone manifolds

TL;DR

This paper demonstrates how to realize all edges in the graph of compression bodies via cone manifold holonomy groups, using CAT(0) space techniques and generalizing classical theorems.

Contribution

It introduces a method to change the topological type of compression bodies through cone manifolds without relying on harmonic deformation theory.

Findings

All edges in the graph of compression bodies can be realized by cone manifold paths.

A generalization of Koebe and Maskit's theorem is proved, ensuring existence of certain hyperbolic cone structures.

Standard CAT(0) space techniques are effectively applied to this topological problem.

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.