Notes on the peripheral volume of hyperbolic 3-manifolds

TL;DR

This paper investigates how much of a hyperbolic 3-manifold's volume can be recovered by boundary collars and cusp neighborhoods, showing maximal volume recovery occurs only in specific extremal configurations.

Contribution

It provides a new analysis of volume recovery in hyperbolic 3-manifolds with boundary or cusps, highlighting extremal configurations for maximal volume.

Findings

Maximum volume recovery occurs only in combinatorially extremal configurations

Elementary techniques are sufficient for the analysis

Results are of interest despite simple methods

Abstract

We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion can only be maximal in some combinatorially extremal configurations. The techniques we employ are VERY elementary but the result is in our opinion of some interest.