Torsion homology growth conjectures for hyperbolic mapping tori

Let $\psi\in\operatorname{Mod}(S)$ be pseudo-Anosov, and let

$$T=T_\psi=S\times I/((s,0)\sim(\psi(s),1))$$

be its mapping torus. A tower of finite covers is a sequence of connected finite covers $\cdots\to T_3\to T_2\to T_1\to T$ whose degrees over $T$ tend to infinity; it is exhausting when the corresponding nested finite-index subgroups have trivial intersection. Torsion homology growth is exponential when the torsion in homology grows exponentially along the tower.

Torsion conjectures. The manifold $T$ has exponential torsion homology growth with respect to some tower of finite covers. In fact, $T$ has exponential torsion homology growth with respect to every tower of exhausting finite covers.

These conjectures connect homological eigenvalues of lifts of pseudo-Anosov mapping classes with torsion growth in finite covers of hyperbolic $3$-manifolds. The source presents both assertions as conjectures and relates the second to work of Bergeron–Venkatesh and Lück.