The length of PU(2,1) relative to special elliptic isometries with fixed parameter

TL;DR

This paper determines that any element of the complex hyperbolic isometry group PU(2,1) can be expressed as a product of at most four special elliptic isometries with a fixed parameter, extending the involution length concept.

Contribution

It establishes the alpha-length of PU(2,1) as 4 and characterizes isometries that are products of 2 or 3 such special elliptic isometries.

Findings

Alpha-length of PU(2,1) is 4

Every element decomposable into 4 special elliptic isometries

Characterization of elements as products of 2 or 3 special elliptic isometries

Abstract

Generalizing the involution length of the complex hyperbolic plane, we obtain that the $\alpha$-length of $\mathrm{PU}(2,1)$ is $4$, that is, every element of $\mathrm{PU}(2,1)$ can be decomposed as the product of at most $4$ special elliptic isometries with parameter $\alpha$. We also describe the isometries that can be written as the product of $2$ or $3$ such special elliptic isometries.