= Cone-torus construction of genus-four Sunada surfaces
{title2=$7\bigl(1-1/7\bigr)=6=2g-2$}
Take a hyperbolic torus with a single cone point of angle $2\pi/7$. Its <orbifold fundamental group> is $\langle a,d\mid[d,a]^7=1\rangle$. An epimorphism to $\mathrm{PSL}(3,2)$ whose commutator has exact order $7$ has torsion-free kernel. The point and line stabilizers have index $7$ and order $24$, hence avoid all nontrivial cone stabilizers. Their quotient covers are smooth surfaces with <Euler characteristic> $7(-6/7)=-6$, so have <genus> $4$. Their equal coset <permutation characters> give equal <heat traces> by the <Sunada theorem>. Nonisometry still requires a geometric distinction, such as the <intersection test for nonisometric finite covers>.
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