Integrable tautness of isometries of complex hyperbolic spaces

Consider $n\ge 2$. In this paper we prove that the group $\text{PU}(n,1)$ is $1$-taut. This result concludes the study of $1$-tautness of rank-one Lie groups of non-compact type. Additionally the tautness property implies a classification of finitely generated groups which are ${\text{L}}^1$-measure equivalent to lattices of $\text{PU}(n,1)$. More precisely, we show that ${\text{L}}^1$-measure equivalent groups must be extensions of lattices of $\text{PU}(n,1)$ by a finite group.