On the Volume Density Spectrum of Fully Augmented Links

TL;DR

This paper investigates the distribution of volume density ratios in hyperbolic fully augmented links, revealing their density and discreteness in specific intervals, which advances understanding of their geometric properties.

Contribution

It characterizes the density spectrum of FAL volume densities, showing they are dense in one interval and discrete in another, a novel insight into their geometric structure.

Findings

FAL volume densities are dense in [2*voct, 10*vtet)

FAL volume densities are discrete in [voct, 2*voct)

The set of densities exhibits a phase transition between density and discreteness.

Abstract

For a hyperbolic fully augmented link in $S^3$, its \emph{FAL volume density} is the ratio of its volume to the number of augmentations. We show that the set of FAL volume densities is dense in $[2\voct, 10\vtet)$, but discrete in $[\voct,2\voct)$.