Let
$$
\Delta(2,3,7)
=\langle A,B,C\mid A^2=B^3=C^7=ABC=1\rangle
$$
be the orientation-preserving $(2,3,7)$ <hyperbolic triangle group>. Define
$$
a\mapsto A,\qquad b\mapsto B,\qquad c\mapsto C,
\qquad x\mapsto1,\qquad y\mapsto1.
$$
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
$$
\frac12+\frac13+\frac17<1,
$$
$\Delta(2,3,7)$ is a non-elementary <Fuchsian group>.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$\Delta(p,q,r)$}
{wiki=Triangle_group}

When $1/p+1/q+1/r<1$, the orientation-preserving hyperbolic triangle group has presentation
$$
\Delta(p,q,r)=\langle a,b,c\mid a^p=b^q=c^r=abc=1\rangle
$$
and is a non-elementary Fuchsian group.

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