Isoperimetric Inequalities using Varopoulos Transport

TL;DR

This paper establishes upper bounds for second order Dehn functions of Nil and Sol groups by applying Varopoulos transport on dual graphs derived from handlebody diagrams, linking geometric and combinatorial methods.

Contribution

It introduces a novel approach using Varopoulos transport on dual graphs to analyze isoperimetric inequalities in three-dimensional groups Nil and Sol.

Findings

Derived upper bounds for second order Dehn functions

Connected handlebody diagrams to dual graph analysis

Applied Varopoulos transport to geometric group theory problems

Abstract

The main results in this paper provide upper bounds of the second order Dehn functions for three-dimensional groups Nil and Sol. These upper bounds are obtained by using the Varopoulos transport argument on dual graphs. The first step is to start with reduced handlebody diagrams of the three-dimensional balls either immersed or embedded in the universal covers of each group and then define dual graphs using the 0-handles as vertices, 1-handles as edges. The idea is to reduce the original isoperimetric problem involving volume of three-dimensional balls and areas of their boundary spheres to a problem involving Varopoulos' notion of volume and boundary of finite domains in dual graphs.