Free isometric actions on the affine space \(\mathbb{Q}^n\).
Let \(G\) be a group of isometries on a metric space \(X\). The action of \(G\) is said to be `locally commutative' if any pair of elements with a common fixed point commutes. For example if \(G\) acts without fixed points its action is locally commutative. This paper is concerned with the action of the special orthogonal group \(\text{SO}_n(\mathbb{Q})\) on the space \(\mathbb{Q}^n\), where \(\mathbb{Q}\) is the set of rational numbers. Using previous results it is proved that, for all \(n \geq 3\), the group \(\text{SO}_n(\mathbb{Q})\) has a subgroup, isomorphic to \(F_2\), which acts on the non-zero elements of \(\mathbb{Q}^n\) in a locally commutative way. It follows that any subgroup of such a group which fixes a non-zero element of \(\mathbb{Q}^n\) is necessarily cyclic. Results of this type have a number of important geometric implications, going back to work of F. Hausdorff. If the group \(F_2\) acts on a space \(X\) in this way, then it often follows that \(X\) can be partitioned by means of subsets which are permuted by elements of the group.
scientific article; zbMATH DE number 1893652