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Unicellular Dessins and a Uniqueness Theorem for Klein's Riemann Surface of Genus 3
Author(s) -
Singerman David
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008347
Subject(s) - mathematics , genus , riemann surface , surface (topology) , automorphism , uniqueness , polygon (computer graphics) , pure mathematics , riemann–hurwitz formula , combinatorics , mathematical analysis , geometric function theory , geometry , telecommunications , frame (networking) , computer science , biology , botany
If we consider the 14‐sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, then we observe that we have a uniform, unifacial dessin whose automorphism group is transitive on the edges, but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property. 2000 Mathematics Subject Classification 30F10, 14H05.