Ebullition in foliated surfaces versus gravitational clumping

TL;DR

This paper explores how the fundamental group influences the complexity of foliated surfaces, revealing phase-like transitions from frozen to gaseous states as the group's rank increases, with implications for non-metric and microscopic surface structures.

Contribution

It provides a novel phase-transition framework for foliated surfaces based on the fundamental group's rank, extending to non-metric and microscopic surface scenarios.

Findings

Surfaces with rank 0-1 groups are intransitive and frozen.

Rank 2-3 surfaces exhibit mixed intransitive and transitive phases.

Rank 4 and above surfaces are transitively foliated, akin to a gaseous phase.

Abstract

For surfaces, we brush a reasonably sharp picture of the influence of the fundamental group upon the complexity of foliated-dynamics. A metaphor emerges with phase-changes through the solid-liquid-gaseous states. Groups of ranks $0\le r\le 1$ are frozen with intransitivity reigning ubiquitously. When $2\le r \le 3$, the marmalade starts its ebullition in the liquid phase, with both regimes (intransitive or not) intermingled after the detailed topology. Whenever $r\ge 4$, we reach the gaseous-volatile phase, where any finitely-connected metric surface is transitively foliated. The game extends non-metrically, as putting to the fridge a frozen configuration keeps it frozen. Gromov asked: {\it Is there a life without a metric?}, yes surely but maybe only a cold eternal one is worth living. The picture is pondered by a scenario of gravitational collapse at the microscopic scale (due to Baillif) dictating the foliated morphology when it comes to surfaces of the Pr\"ufer type (like those of R.\,L. Moore or Calabi-Rosenlicht).