## ::(2,3,7) triangle group

### ::concepts

Group::triangle Tiling::algebra Order::surfaces Riemann::category Hurwitz::which Fuchsian::schwarz

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus *g* with the largest possible order, 84(*g* − 1), of its automorphism group.

A note on terminology – the "(2,3,7) triangle group" most often refers, not to the *full* triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the *ordinary* triangle group (the von Dyck group) *D*(2,3,7) of orientation-preserving maps (the rotation group), which is index 2.

Torsion-free normal subgroups of the (2,3,7) triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet.

**(2,3,7) triangle group sections**

Intro Constructions Relation to SL(2,R) References Further reading

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