We study locally flat, compact, oriented surfaces in $4$--manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we provide a classification of a subset of the topological $4$--manifolds with infinite cyclic fundamental group, and we apply our results to rim surgery.
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