All finitely presentable groups from link complements and Kleinian groups

TL;DR

This paper demonstrates that every finitely presentable group can be realized as the fundamental group of a modified hyperbolic link complement, introducing the closed-link-genus to classify such groups within 3-manifold topology.

Contribution

It introduces the closed-link-genus as a new invariant that characterizes fundamental groups of closed orientable 3-manifolds and relates it to existing concepts like genus(G).

Findings

Every finitely presentable group arises from a hyperbolic link complement.

The closed-link-genus fully characterizes fundamental groups of closed orientable 3-manifolds.

clg(G) bounds the genus(G) and relates to minimal relations in group presentations.

Abstract

We prove that every finitely presentable group G arises as the fundamental group of an orientable 3-complex obtained from a hyperbolic link complement, by coning each boundary torus of the link exterior to a distinct point. We define the closed-link-genus, clg(G), of any finitely presentable group G, which completely characterizes fundamental groups of closed orientable 3-manifolds: clg(G)=0 if and only if G is the fundamental group of a closed orientable 3-manifold. Moreover clg(G) gives an upper bound for the concept `genus(G)' of genus defined earlier by Aitchison and Reeves, and in turn is bounded by the minimal number of relations among all finite presentations of G.