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

TL;DR

This paper investigates surface-group representations into the complex hyperbolic group PU(2,1), establishing domination results for a special class called T-bent representations and linking them to discrete, faithful real hyperbolic representations.

Contribution

It introduces domination results for T-bent representations into PU(2,1), connecting them to discrete, faithful representations into PO(2,1) that preserve peripheral lengths.

Findings

T-bent representations are dominated by discrete, faithful PO(2,1) representations.

Domination preserves peripheral loop lengths.

Results apply to punctured surfaces with negative Euler characteristic.

Abstract

This article explores surface-group representations into the complex hyperbolic group $\mathrm{PU}(2,1)$ 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 $\rho: \pi_1(S_{g,k}) \rightarrow \mathrm{PU}(2,1)$, there exists a discrete and faithful representation $\rho_0: \pi_1(S_{g,k}) \rightarrow \mathrm{PO}(2,1)$ that dominates $\rho$ in the Bergman translation length spectrum, while preserving the lengths of the peripheral loops.