All finitely presentable groups from link complements and Kleinian groups

Details All finitely presentable groups from link complements and Kleinian groups All finitely presentable groups from link complements and Kleinian groups

2010-08-07

arxiv.org/abs/1008.1311v1

Mathematics

Geometric Topology

18 pages, 4 figures

Scientific paper

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.

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