Systole–Pachner growth conjecture for cusped hyperbolic 3-manifolds

Establish a quantitative relation between the growth rate of the Pachner graph of a cusped hyperbolic 3-manifold and the inverse of its systole, i.e., the length of its shortest closed geodesic; in particular, formalize and prove the observed inverse correlation between these quantities.

Background

For 4,815 orientable cusped hyperbolic 3-manifolds from the SnapPy census (with fewer than eight initial tetrahedra), the authors generated Pachner graphs (up to depth 3) using both 2-3/3-2 and 1-4/4-1 moves and performed large-scale network analysis.

Scatter plots of graph sizes against geometric data showed that larger Pachner graphs tend to correspond to shorter systoles, and families of points aligned according to the drilled manifolds. These empirical patterns led the authors to formulate a conjecture linking Pachner graph growth to the inverse of the systole length.

References

Conjecture. The growth rate of the Pachner graph of a cusped hyperbolic $3$-manifold $M$ is correlated to the inverse of the length of the shortest geodesic in $M$.

— Learning 3-Manifold Triangulations (2405.09610 - Costantino et al., 15 May 2024) in Conjecture 1, Subsubsection “Orientable Cusped Census”