Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.

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:

a

2

=

b

2

=

c

2

=

1

{\displaystyle a^{2}=b^{2}=c^{2}=1}

and

(

a

b

)

l

=

(

b

c

)

n

=

(

c

a

)

m

=

1.

{\displaystyle (ab)^{l}=(bc)^{n}=(ca)^{m}=1.}

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

Δ

(

l

,

m

,

n

)

=

⟨

a

,

b

,

c

∣

a

2

=

b

2

=

c

2

=

(

a

b

)

l

=

(

b

c

)

n

=

(

c

a

)

m

=

1

⟩

.

{\displaystyle \Delta (l,m,n)=\langle a,b,c\mid a^{2}=b^{2}=c^{2}=(ab)^{l}=(bc)^{n}=(ca)^{m}=1\rangle .}

An abstract group with this presentation is a Coxeter group with three generators.