Cofinal towers with vanishing homology torsion

Abstract

Problem 3.6 in the $\mathrm{K3}$ problem list of Baykur, Kirby and Ruberman asks whether every cofinal tower \[ M_0\longleftarrow M_1\longleftarrow M_2\longleftarrow\cdots \] of finite covers of a finite-volume hyperbolic $3$-manifold satisfies \[ \lim_{n\to\infty} \frac{\log|\operatorname{Tor} H_1(M_n;\mathbb Z)|} {\operatorname{vol}(M_n)} =\frac{1}{6\pi}. \] We give a negative answer. For every ideal right-angled polyhedron $P_0$, the checkerboard manifold associated to $P_0$ admits a cofinal tower all of whose levels are hyperbolic link complements in $S^3$. Thus $\operatorname{Tor} H_1(M_n;\mathbb Z)=0$ at every level, and the normalized logarithmic homology torsion is identically zero.