Kirby Problem 3.6

Definitions and mathematical statement

Let $M=\mathbb H^3/\Gamma$ be a connected closed hyperbolic three-manifold, with curvature $-1$, and let $\Gamma\geq\Gamma_1\geq\Gamma_2\geq\cdots$ be finite-index subgroups. They define a tower of connected finite covers $M_n=\mathbb H^3/\Gamma_n$. The tower is cofinal if $\bigcap_n\Gamma_n=\{1\}$. Let $T_n=\operatorname{Tor}H_1(M_n;\mathbb Z)$, the finite subgroup of homology classes of finite order, and put $|T_n|=1$ when it is trivial.

The displayed question in the catalogue is whether \[ \lim_{n\to\infty} \frac{\log|T_n|}{\operatorname{vol}(M_n)}=\frac1{6\pi}, \qquad \operatorname{vol}(M_n)=[\Gamma:\Gamma_n]\operatorname{vol}(M), \] where $\log$ denotes the natural logarithm. The hypothesis on the tower is substantive. The source distinguishes the arithmetic congruence-tower conjecture from the nonarithmetic formulation using regular covers and $b_1(M_n)=\dim_{\mathbb Q}H_1(M_n;\mathbb Q)=0$ for every $n$. Regular means $\Gamma_n$ is normal in $\Gamma$; congruence means that the subgroups arise from finite congruence quotients of the arithmetic lattice. For arbitrary cofinal towers the same equality is posed as a broader question, with source evidence warning that positive Betti numbers can change the behavior. These additional hypotheses must not be silently dropped from the restricted conjectures.

Short English statement

Determine when torsion in first homology grows at the universal exponential rate predicted by hyperbolic volume along a cofinal tower of finite covers.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2804 (KP-3.6), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.6. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.