= Triangle cover construction for Sunada surfaces
{title2=$\chi(M)=|T|(p^{-1}+q^{-1}+r^{-1}-1)$}
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>.
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