Abstract
For any g ≥ 2 we construct a graph Γg ⊂ S3 whose exterior Mg = S3\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 self-homeomorphism of Mg extends to a self-homeomorphism of the pair (S3,Γg), and that Γg is chiral. Building on a result of Lackenby we also show that any non-meridinal Dehn filling of Mg is hyperbolic, thus getting an infinite family of graphs in S2 × S1 whose exteriors support a hyperbolic structure with geodesic boundary.
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