Torsion-homology conjecture for strongly isospectral hyperbolic 3-manifolds

Let $p$ be a prime number. A closed hyperbolic $3$-manifold is a closed $3$-manifold equipped with a hyperbolic metric, and two Riemannian manifolds are strongly isospectral when they have identical spectra for all natural differential operators on natural vector bundles. For a finitely generated abelian group $H$, let $H[p^\infty]$ denote its $p$-primary torsion subgroup, and let $\#H[p^\infty]$ denote its order.

Torsion-homology conjecture. For every prime number $p$, there exist strongly isospectral closed hyperbolic $3$-manifolds $M_1$ and $M_2$ such that

$$\#H_1(M_1,\Z)[p^\infty]\neq \#H_1(M_2,\Z)[p^\infty].$$

The proposition preceding the conjecture proves the assertion for every prime $p\leq 71$, providing evidence that strongly isospectral hyperbolic $3$-manifolds can have different $p$-primary torsion in their first homology. The paper notes that the conjecture was subsequently established in later work using techniques introduced here.