Li-Yau inequality on virtually Abelian groups

TL;DR

This paper demonstrates that Cayley graphs of virtually Abelian groups satisfy a Li-Yau gradient estimate, revealing new geometric properties despite lacking known curvature-dimension inequalities.

Contribution

It establishes a Li-Yau inequality for virtually Abelian groups' Cayley graphs, expanding understanding beyond traditional curvature-dimension conditions.

Findings

Cayley graphs of virtually Abelian groups satisfy Li-Yau gradient estimates

These graphs do not meet known curvature-dimension inequalities

The result broadens the class of graphs known to satisfy Li-Yau inequalities

Abstract

We show that Cayley graphs of virtually Abelian groups satisfy a Li-Yau type gradient estimate despite the fact that they do not satisfy any known variant of the curvature-dimension inequality with non-negative curvature.