 
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 EpsteinPennerFord 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 "8fold 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
1skeleton is the complete graph on 7 vertices. Thurston wondered if
