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THURSTON'S CONGRUENCE LINK Klein's quartic curve may be described as the Riemann surface ob-
 

Summary: THURSTON'S CONGRUENCE LINK
IAN AGOL
Klein's quartic curve may be described as the Riemann surface ob-
tained by taking the quotient of H2
by the (principal congruence) sub-
group (7) = ker {PSL2(Z) PSL2(Z/7Z)}, and filling points in the
cusps (punctures) to get a closed surface (although the punctured sur-
face is sometimes also referred to as Klein's quartic). It has a cell
decomposition by 24 heptagons, centered at each cusp coming from
the Epstein-Penner-Ford domain of H2
/(7). Each heptagon is fixed
by a rotation of order 7, which also preserves two other heptagons, giv-
ing a grouping of the heptagons into 8 classes which are preserved by
the symmetries of the surface. Rotating one heptagon 1/7th of a turn
corresponds to rotating one other 2/7ths, and the third 4/7ths. During
a lecture at MSRI on Klein's quartic commemorating the installation
of Helaman Ferguson's sculpture "8-fold way" [2], Thurston noticed
that the group of symmetries preserving each class of heptagons is the
same as the group of symmetries of the triangulation of the torus whose
1-skeleton is the complete graph on 7 vertices. Thurston wondered if

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics