Triangle cover construction for Sunada surfaces ($\chi (M)=|T|(p^{-1}+q^{-1}+r^{-1}-1)$)

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.

Write triangle-group generators as products $a=s_1{s}_2$, $b=s_2{s}_3$ of adjacent side reflections. Conjugation by $s_2$ sends both to their inverses. A sheet relabelling satisfying the displayed two permutation identities therefore lifts this base reflection to an orientation-reversing isometry between the two hyperbolic covers. Thus nonconjugate subgroups in the orientation-preserving finite group can still give isometric covers.