Revista de la
Unión Matemática Argentina
Generalizations of hyperbolic area for topological surfaces
Aldo-Hilario Cruz-Cota
Volume 59, no. 2 (2018), pp. 431–441


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We introduce two generalizations of hyperbolic area for connected, closed, orientable surfaces: the complexity and the simple complexity of a surface. These concepts are defined in terms of collections of branched coverings $M \to \mathbb{P} ^1$, where $M$ is a Riemann surface homeomorphic to $S$ and $ \mathbb{P} ^1$ is the Riemann sphere. We prove that if $S$ is a surface of positive genus, then both the topological complexity and the simple topological complexity of $S$ are linear functions of its genus.