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If a graph is in bridge position in a 3-manifold so that the graph complementis irreducible and boundary irreducible, we generalize a result of Bachman andSchleimer to prove that the complexity of a surface properly embedded in thecomplement of the graph bounds the graph distance of the bridge surface. We usethis result to construct, for any natural number $n$, a hyperbolic manifoldcontaining a surface of topological index $n$.

Hyperbolic manifolds containing high topological index surfaces