Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 19 / 5 / 2 / Solution

For each $k$, the lifted Riemannian metric on $X$ is $T$-invariant and the two subgroups still act freely. Sunada's theorem therefore givesTo rule out any other equalities, the leading diagonal heat kernel expansion from Question 3 gives, on a closed surface,Thus the function spectrum determines area: multiply the heat trace by $4\pi t$ and take the limit. Since the chosen areas are distinct for distinct $k$, no cross-pair spectra can agree. This proves exactly the prescribed spectral coincidences.