# Free affine $\mathbb{Z}^p$-actions on Tori

## TL;DR

This paper characterizes when certain automorphism actions of f6p on tori are realizable as free affine actions, providing criteria based on fixed-point sets and classifying actions on 3-dimensional tori.

## Contribution

It introduces criteria for when f6p-actions are liberated affine, especially for unipotent actions, and classifies actions on 3-dimensional tori.

## Key findings

- Unipotent f6p-actions with small fixed-point sets are liberated affine.
- All actions on f6p with f6pb4s fixed-point set dimension f6q/2 are liberated affine.
- Complete classification of unipotent f6p-actions on f6q=3 as linear parts of free affine actions.

## Abstract

We prove that any $\mathbb{Z}^p$-action ${\bf A}$ that acts by automorphisms of $\mathbb{Z}^q$ with a non-zero fixed-point set induces a unipotent factor of the $\mathbb{Z}^p$-action ${\bf A}$ which determines whether the action ${\bf A}$ is {\it liberated affine}, i.e. ${\bf A}$ is the linear part of a free affine $\mathbb{Z}^p$-action on the torus $T^q$. In general, it is not true that all unipotent $\mathbb{Z}^p$-actions $\bf{U}$ on $\mathbb{Z}^q$ are liberated affine: counter-examples appear for $q\geq 4$. But if the dimension of the fixed-point set of ${\bf U}$ regarded as a subspace of $\mathbb{Z}^q$ is less than $q/2$, then ${\bf U}$ is liberated affine. If $q\leq3$, then a $\mathbb{Z}^p$-action on $\mathbb{Z}^q$ with non-zero fixed-point set is liberated affine. Finally, for unipotent $\mathbb{Z}^p$-actions on $\mathbb{Z}^3$, we obtain a classification of all those that are the linear part of a minimal free affine $\mathbb{Z}^p$-action on $T^3$.