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

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.