A free group acting on \(\mathbb{Z}^2\) without fixed points
Let \(\text{SA}_2(\mathbb{Z})\) be the group of all transformations of \(\mathbb{Z}^2\) of the type \({\mathbf x}\mapsto A{\mathbf x}+{\mathbf a}\), where \(A\in\text{SL}_2(\mathbb{Z})\) and \({\mathbf a}\in\mathbb{Z}^2\). The author proves that \(\text{SA}_2(\mathbb{Z})\) contains a free group of rank 2 which acts on \(\mathbb{Z}^2\) without non-trivial fixed points. This is related to the problem of determining whether or not the more general group \(\text{SA}_2(\mathbb{R})\), which acts on \(\mathbb{R}^2\), contains a free subgroup \(F\) of rank 2, with the property that the stabilizer in \(F\) of every point of \(\mathbb{R}^2\) is commutative. Questions like these arise from the so-called Banach-Tarski paradox for bounded subsets of \(\mathbb{R}^3\).
scientific article; zbMATH DE number 1558734