# Hyperbolic manifolds containing high topological index surfaces

## TL;DR

This paper establishes a relationship between the complexity of embedded surfaces in 3-manifolds with graphs and the graph distance of bridge surfaces, and constructs hyperbolic manifolds with surfaces of arbitrary topological index.

## Contribution

It generalizes previous results to include graphs in 3-manifolds and constructs hyperbolic manifolds with surfaces of any given topological index.

## Key findings

- Surface complexity bounds graph distance in bridge position
- Constructs hyperbolic manifolds with surfaces of arbitrary topological index
- Extends previous work to graph complements in 3-manifolds

## 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 $n$.