In this paper an overview of the tools and techniques needed for successfully classifying “low-complexity” Kleinian groups is given. In particular the manuscript is focused on extracting topological and geometric properties of discrete Kleinian groups, such as bounds on tube radii, cusp geometry, volume, relators in group presentation and similar quantities. A key point of this work is to explain how a discrete set of solutions (or of their closure) can be found using continuous methods, in particular by searching over a continuous parameter space of groups. The parameter space \(\mathcal{P}\subset \mathbb{C}^{d}\) is build to be large enough in order to include parameters representing each interesting equivalence class of markings that arise from a hyperbolic manifold. As a result, the goal of the paper is to describe methods to eliminate places of the parameter space where some geometric measurement violates discreteness or other assumptions. Thus, the hard work of the paper is to understand how to cut away “bad” portions of the parameter space where, for example, discrete groups cannot exist. At this point the generators of the groups play an essential role. In the paper many interesting examples which focus on 2 or 3 generator subgroups are presented, as their parameter space is small enough to apply computational methods. Generally, the present work provides effective methods for studying and classifying hyperbolic 3-manifolds that satisfy some geometric or topological constraints.
For the entire collection see [Zbl 1511.20004].