This paper contains two main results. The first one (Theorem 1.2) is one of the numerous generalisations of a result of K. Hartshorn [Pac. J. Math. 204, No. 1, 61–75 (2002; Zbl 1065.57021)]. It is a technical result relating the complexity of an essential surface \(S\) in a closed \(3\)-manifold \(M\) and its graph bridge distance. More precisely, it states that the graph distance of a graph that is in bridge position with respect to a Heegaard surface of \(M\) is bounded above by \(2(2g(S) + |\partial S| - 1)\), where \(g(S)\) is the genus of \(S\). This Theorem is a key ingredient in the proof of the second main result of the paper (Theorem 1.1). The authors generalise the construction in [D. Bachman and J. Johnson, Math. Res. Lett. 17, No. 3, 389–394 (2010; Zbl 1257.57026)] to show that there is a closed \(3\)-manifold \(M^1\), with an index \(1\) Heegaard surface \(S\), such that for each \(n\), the lift of \(S\) to some \(n\)-fold cover \(M^n\) of \(M^1\) has topological index \(n\). Moreover, \(M^n\) is hyperbolic for all \(n\). The construction of \(M^n\) is very explicit. The authors take a special two component link in the three sphere and modify it by adding arcs. Then they remove tubular neighbourhoods of the two resulting graphs to obtain a three manifold \(M'\) with two boundary components. The manifold \(M^n\) is the closed manifold generated by gluing \(n\) copies of \(M'\) by pairing their boundaries.