#### Vol. 296, No. 2, 2018

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Hyperbolic manifolds containing high topological index surfaces

### Marion Campisi and Matt Rathbun

Vol. 296 (2018), No. 2, 305–319
##### Abstract

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

##### Keywords
topological index, topologically minimal, hyperbolic, bridge position, distance, bridge distance, graph
##### Mathematical Subject Classification 2010
Primary: 55P15, 57M20, 57M27
Secondary: 57M10, 57M15