A note on the Hurwitz problem and cone spherical metrics

2022/10/18 by Song, Jijian, Xu, Bin, Ye, Yu
#20B35 #30F30 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2210.09700

Abstract

We are motivated by cone spherical metrics on compact Riemann surfaces of positive genus to solve a special case of the Hurwitz problem. Precisely speaking, letting d, g and ℓ be three positive integers and Λ be the following collection of (ℓ+2) partitions of a positive integer d: (a₁,⋯, a_p), (b₁,⋯, b_q), (m₁+1,1,⋯,1),⋯, (m_ℓ+1,1,⋯,1), where (m₁,⋯, m_ℓ) is a partition of p+q-2+2g, we prove that there exists a branched cover from some compact Riemann surface of genus g to the Riemann sphere \Bbb P¹ with branch data Λ. An analogue for the genus-zero case was found by the first two authors (\it Algebra Colloq. \bf 27 (2020), no. 2, 231-246), who were stimulated by such metrics on \Bbb P¹ and conjectured the veracity of the above statement there.

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