# Extensions of Abelian Automata Groups

## TL;DR

This paper investigates how the choice of residuation vector influences the structure of Abelian automata groups, revealing that all variations extend from an initial standard basis vector case by adding fractional elements.

## Contribution

It characterizes the impact of the residuation vector on Abelian automata groups, showing how different choices extend a fundamental initial structure.

## Key findings

- Identifies the initial structure when e is the first standard basis vector
- Shows how other choices of e extend the initial structure with fractional elements
- Provides a precise description of the extension process

## Abstract

A theorem of Nekrashevych and Sidki shows the Mealy Automata structures one can place on Z^m are parametrized by a family of matrices (called "1/2-integral") and a choice of residuation vector e in Z^m. While the impact of the chosen matrix is well understood, the impact of the residuation vector on the resulting structure is seemingly sporadic. In this paper we characterize the impact of the residuation vector e by recognizing an initial structure when e is the first standard basis vector. All other choices of e extend this initial structure by adding "fractional elements" in a way we make precise.