Definition
Let l, m, n be integers greater than or equal to 2. A triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/l, 2π/m and 2π/n Therefore, if the generating reflections are labeled a, b, c and the angles between them in the cyclic order are as given above, then the following relations hold:
It is a theorem that all other relations between a, b, c are consequences of these relations and that Δ(l,m,n) is a discrete group of motions of the corresponding space. Thus a triangle group is a reflection group that admits a group presentation
An abstract group with this presentation is a Coxeter group with three generators.
Read more about this topic: Triangle Group
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