Convex Hull Volumes in Hyperbolic 3-Space

TL;DR

This paper characterizes when the hyperbolic convex hulls of certain closed sets in the Riemann sphere have infinite volume, providing geometric conditions and answering a question by Calegari.

Contribution

It offers a new geometric condition for infinite volume convex hulls in hyperbolic 3-space and characterizes specific sets with this property.

Findings

Characterization of continua with infinite hyperbolic convex hull volume

Geometric condition for infinite volume in hyperbolic convex hulls

Answer to Calegari's question about convex hulls

Abstract

In this paper we provide a geometric condition satisfied by certain closed subsets of the Riemann sphere which implies that their hyperbolic convex hulls in $\mathbb{H}^3$ have infinite volume. As a corollary, we characterize continua in the Riemann sphere whose hyperbolic convex hulls have infinite volume, answering a question of Danny Calegari. Furthermore, we give a geometric characterization of planar self-similar sets whose hyperbolic convex hulls have infinite volume.