# Lack of Sphere Packing of Graphs via Non-Linear Potential Theory

## TL;DR

This paper proves the impossibility of certain sphere packings in Euclidean space for specific infinite graphs using non-linear potential theory, highlighting limitations in geometric embeddings.

## Contribution

It introduces a novel non-linear potential theory approach to establish non-existence results for sphere packings of various infinite graphs.

## Key findings

- No quasi-sphere packing of Z^{d+1} in R^d.
- No quasi-sphere packing of hyperbolic lattices in R^d.
- Non-existence results for some other graphs.

## Abstract

It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from non-linear potential theory.