On the domination of surface-group representations in $\mathrm{P}\mathrm{U}(2,1)$

This article explores surface-group representations into the complex hyperbolic group $\mathrm{P}\mathrm{U}(2,1)$ and presents domination results for a special class of representations called $T$-bent representations. Let $S_{g,k}$ be a punctured surface of negative Euler characteristic. We prove that for a $T$-bent representation $\rho :{\pi }_1(S_{g,k})\to \mathrm{P}\mathrm{U}(2,1)$, there exists a discrete and faithful representation ${\rho }_0:{\pi }_1(S_{g,k})\to \mathrm{P}\mathrm{O}(2,1)$ that dominates $\rho$ in the Bergman translation length spectrum, while preserving the lengths of the peripheral loops.