# Higher rank lamplighter groups are graph automatic

## TL;DR

This paper proves that higher rank lamplighter groups, specifically Diestel-Leader groups for dimensions three and above, are graph automatic, expanding the class of known graph automatic groups beyond traditional automatic groups.

## Contribution

It demonstrates that Diestel-Leader groups in higher dimensions are graph automatic, introducing a new family of groups with this property that are not automatic.

## Key findings

- Higher rank lamplighter groups are graph automatic.
- These groups are not automatic, showing a broader class of graph automatic groups.
- The result applies to Diestel-Leader groups for all dimensions d ≥ 3.

## Abstract

We show that the higher rank lamplighter groups, or Diestel-Leader groups $\Gamma_d(q)$ for $d \geq 3$, are graph automatic. This introduces a new family of graph automatic groups which are not automatic.