An infinite family of hyperbolic graph complements in S 3

For any g > 2 we construct a graph Γg ⊂ S whose exteriorMg = S\N(Γg) supports a complete finite-volume hyperbolic structure with one toric cusp and a connected geodesic boundary of genus g. We compute the canonical decomposition and the isometry group of Mg, showing in particular that any selfhomeomorphism of Mg extends to a self-homeomorphism of the pair (S ,Γg), and that Γg is chiral. Building on a result of Lackenby [5] we also show that any non-meridinal Dehn filling of Mg is hyperbolic, thus getting an infinite family of graphs in S ×S1 whose exteriors support a hyperbolic structure with geodesic boundary. MSC (2000): 57M50 (primary), 57M15 (secondary). 1 Preliminaries and statements In this paper we introduce an infinite class {Γg, g > 2} of graphs in S3 whose exteriors support a complete finite-volume hyperbolic structure with geodesic boundary. Any Γg has two connected components, one of which is a knot. We describe some geometric and topological properties of the Γg’s, and we show that for any g > 2 any non-meridinal Dehn-filling of the torus boundary of the exterior of Γg gives a compact hyperbolic manifold with geodesic boundary. Definition of Γg and hyperbolicity We say that a compact orientable 3-manifold is hyperbolic if, after removing the boundary tori, we get a complete finite-volume hyperbolic 3-manifold with geodesic boundary. Let Γ be a graph in a closed 3manifold M and let N(Γ) ⊂ M be an open regular neighbourhood of Γ in M . We say that Γ is hyperbolic if M \N(Γ) is hyperbolic. If so, Mostow-Prasad’s Rigidity Theorem (see [3, 2] for a proof in the case with non-empty geodesic boundary) ensures that the complete finite-volume hyperbolic structure with geodesic boundary on M \N(Γ) is unique up to isometry.