Hyperbolic vs non-hyperbolic Pachner graph growth for closed 3-manifolds

Determine whether the Pachner graphs of closed hyperbolic 3-manifolds grow much faster, as a function of move depth (e.g., the size of balls of given radius), than the Pachner graphs of non-hyperbolic closed 3-manifolds.

Background

The paper constructs 1-vertex Pachner graphs for several closed 3-manifolds using sequences of 2-3 and 3-2 moves, and studies their growth with depth. Empirically, the Weeks manifold and other hyperbolic examples exhibited substantially faster growth, hitting computational limits at smaller depths than non-hyperbolic manifolds.

Motivated by these observations, the authors explicitly conjecture a general phenomenon that, for closed 3-manifolds, hyperbolic geometry entails much faster growth of the Pachner graph compared to non-hyperbolic geometries. Formalizing and proving this comparative growth behavior would connect geometric type to combinatorial triangulation dynamics.

References

Also, the Weeks manifold (and other hyperbolic manifolds not of focus here) exhibited significantly faster Pachner graph growth with depth, reaching computational limits and leading us to conjecture that Pachner graphs of closed hyperbolic $3$-manifolds grow much faster than those of non-hyperbolic $3$-manifolds.

— Learning 3-Manifold Triangulations (2405.09610 - Costantino et al., 15 May 2024) in Introduction (paragraph summarizing network analysis observations)