| An (abstract) tropical curve is a finite metric graph (possibly with further decorations such as integer vertex weights or a sheaf of harmonic functions). The topological fundamental group of the underlying graph is a finitely generated free group that classifies all topological covers. One might suspect that this is all there is. In this talk I intend to convince you that there are at least two other different candidates that answer the question in the title: one that classifies tropical admissible covers, and another that classifies realizable tropical admissible covers. This gives a new perspective on the classical correspondence theorem for algebraic and tropical Hurwitz numbers and allows us to (re-)construct tropical/logarithmic compactifications of the moduli space of curves with level structures and of profinite Teichmueller space. This talk surveys joint work in progress with Yoav Len, Mattia Talpo, and Dmitry Zakharov. |