All finitely presentable groups from link complements and Kleinian groups

Klein defined geometry in terms of invariance under groups actions; here we give a discrete (partial) converse of this, interpreting all (finitely presentable) groups in terms of the geometry of hyperbolic 3-manifolds (whose fundamental groups are, appropriately, Kleinian groups). For G∗ a Kleinian group of isometries of hyperbolic 3-space H, with MG∗ ∼= H3/G∗ a non-compact N -cusped orientable 3-manifold of finite volume, let PG∗ ⊂ S ∞ = ∂H̄ be its dense set of parabolic fixed points. Let M̄G∗ := H ∪PG∗/G be the 3-complex obtained by compactifying each cusp of MG∗ with an additional point. This is the 3-dimensional analogue of the standard compactifcation of cusps of hyperbolic Riemann surfaces. We prove that every finitely presentable group G is of the form G = π1(M̄G∗), in infinitely many ways: thus every finitely presentable group arises as the fundamental group of an orientable 3-complex M̄ – denoted as a ‘link-singular’ 3-manifold – 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. Our results place some aspects of the study of finitely presentable groups more centrally within both classical and modern 3-manifold topology: accordingly, proofs given are expressed in these terms, although some can be seen naturally in 4-manifold topology. 2010 MSC. Primary: 57M05, 57N10, 30F40; Secondary: 20F05, 20F65, 57M50