Asymptotic minimum systole for bounded-tetrahedron manifolds

Establish that for the set \(\Omega_n\) of orientable hyperbolic manifolds admitting ideal triangulations with at most \(n\) tetrahedra, \(\lim_{n\to\infty}\min_{M\in\Omega_n}\systole(M)\phi^{2n}=1.266\dots\), and prove that for every \(n\geq2\) the minimum is uniquely achieved by \(\mathrm{m125}(0,0)(F_n,F_{n+1})\) with the specified Fibonacci layered-torus triangulation and canonical cell decomposition.

Background

The authors tabulate the shortest systoles among census manifolds with two through eleven tetrahedra. The observed manifolds arise from Fibonacci-patterned layered solid tori attached to the triangulation cHcbbdh.

The conjecture proposes both an exact asymptotic decay rate for the minimum systole and a precise extremal classification for every tetrahedron bound. The paper gives heuristic support based on the growth of coefficients in alternating layered solid tori but does not prove the claim.

References

These data lead us to the following conjecture:

Let $\Omega_n$ be the set of orientable hyperbolic manifolds which decompose into $n$ or fewer ideal tetrahedra. Let $F_1=1, F_2=1, F_{n}=F_{n-1}+F_{n-2}$ be the Fibonacci numbers. Then, $$ \lim_{n\to\infty} \min_{M \in \Omega_n} \systole(M) \cdot \phi{2n} = 1.266\dots$$ where $\systole(M)$ is the systole of $M$ and $\phi=\big(\,1+\sqrt{5}\,\big)\big/2$. Furthermore, for $n\geq 2$, the minimum is achieved by the manifold m125(0,0)($F_n$,$F_{n+1}$) which has an ideal triangulation with $n$ tetrahedra obtained from the triangulation cHcbbdh by joining the two unglued faces (for $n=2$) or attaching a solid layered torus $(F_{n-1},F_n, F_{n+1})$ (for $n>2$). This triangulation is also the unique minimal ideal geometric triangulation and the canonical cell decomposition of m125(0,0)($F_n$,$F_{n+1}$).

— Verified Length Spectrum and Margulis number for Hyperbolic 3-Manifolds  (2609.26663 - Goerner et al., 22 Sep 2026) in Introduction, Conjecture 1 (following Table 1)