Space form isometries as commutators and products of involutions

Abstract:

In dimensions 2 and 3 it is well known that given two orientation-preserving hyperbolic isometries that generate a non-elementary group, one can find a triple of involutions so that each isometry can be expressed as a product of two of the three involutions; in this case, we say that the isometries are linked.

In this paper, we investigate the extent to which a pair of isometries in higher dimensions can be linked. This question separates naturally into two parts. In the first part, we determine the least number of involutions needed to express an isometry as a product, and give two applications of our results; the second part is devoted to the question of linking.

In general, the commutator (involution) length of a group element is the least number of elements needed to express that element as a product of commutators (involutions), and the commutator (involution) length of the group is the supremum over all commutator (involution) lengths. Let ${\mathcal G}^n$ be the group of orientation-preserving isometries of one of the space forms, the $(n - 1)$-sphere, Euclidean $n$-space, hyperbolic $n$-space. For $n \geq 3$, we show that the commutator length of ${\mathcal G}^n$ is 1; i.e., every element of ${\mathcal G}^n$ is a commutator. We also show that every element of ${\mathcal G}^n$ can be written as a product of two involutions, not necessarily orientation-preserving; and, depending on the particular space and on the congruence class of $\;n\!\mod 4$, the involution length of ${\mathcal G}^n$ is either 2 or 3.

In the second part of the paper, we show that all pairs in $SO^4$ are linked but that the generic pair in the orientation-preserving isometries of hyperbolic 4-space or in ${\mathcal G}^n$, $n \geq 5$, is not.