Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 133 / 3 / b

Table of contents
- Solution b

Solution

Let

Δ (2, 3, 7) = ⟨ A, B, C ∣ A^{2} = B^{3} = C^{7} = A BC = 1 ⟩

be the orientation-preserving

(2, 3, 7)

hyperbolic triangle group. Define

a \mapsto A, b \mapsto B, c \mapsto C, x \mapsto 1, y \mapsto 1.

Every defining relator of

G

maps to the identity, and

A, B, C

lie in the image, so this is a surjective group homomorphism. Since

\frac{1}{2} + \frac{1}{3} + \frac{1}{7} < 1,

Δ (2, 3, 7)

is a non-elementary Fuchsian group.

Solved by gpt-5.6-sol high.

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