 Triangle cover construction for Sunada surfaces

Let a finite group $T$ be generated by elements of exact orders $p,q$ whose product has exact order $r$, with $p^{-1}+q^{-1}+r^{-1}<1$. The epimorphism from the hyperbolic triangle group sending its generators to these elements has torsion-free kernel. Its quotient of the hyperbolic plane is a closed surface with a $T$ action. A subgroup acts freely precisely when it meets no conjugate of a nonidentity power of the three elliptic generators. Free almost conjugate subgroups give isospectral manifolds by the Sunada theorem.