Geodesic
From Farey fractions to the Klein quartic and beyond
Ars Mathematica Contemporanea, Tome 20 (2021) no. 1, pp. 37-50.

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In a paper published in 1878/79 Klein produced his famous 14-sided polygon representing the Klein quartic, his Riemann surface of genus 3 which has PSL(2, 7) as its automorphism group. The construction and method of side pairings are fairly complicated. By considering the Farey map modulo 7 we show how to obtain a fundamental polygon for Klein’s surface using arithmetic. Now the side pairings are immediate and essentially the same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in a follow-up paper of 1879.
DOI : 10.26493/1855-3974.2046.cb6
Keywords: Riemann surfaces, Klein quartic, regular maps, Farey tessellation, modular group, principal congruence subgroups 
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Ioannis Ivrissimtzis; David Singerman; James Strudwick. From Farey fractions to the Klein quartic and beyond. Ars Mathematica Contemporanea, Tome 20 (2021) no. 1, pp. 37-50. doi : 10.26493/1855-3974.2046.cb6. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2046.cb6/

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