2011/10/28 by Aldo-Hilario Cruz-Cota, Cruz-Cota, Aldo-Hilario, Teresita Ramirez-Rosas +1
Computer Science · Mathematics · #57M12 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:57M12 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1110.6453
9 pages
arxiv created 2011/10/28 · openalex publication_date 2011/10/28 · arxiv updated 2011/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
\noindent Given a Riemann surface M, the complexity of a branched cover of M to the Riemann sphere S2, of degree d and with branching set of cardinality n ≥ 3, is defined as d times the hyperbolic area of the complement of its branching set in S2. A branched cover p \colon M → S2 of degree d is simple if the cardinality of the pre-image p-1(y) is at least d-1 for all y ∈ S2. The (simple) complexity of M is defined as the infimum of the complexities of all (simple) branched covers of M to S2. We prove that if M is a closed, connected, orientable Riemann surface of genus g ≥ 1, then: (1) its simple complexity equals 8πg, and (2) its complexity equals 2π(mmin+2g-2), where mmin is the minimum total length of a branch datum realizable by a branched cover p \colon M → S2.