For any g > 2 we construct a graph Γg ⊂ S whose exterior Mg = 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… CONTINUE READING