Rigidity at infinity for lattices in rank-one Lie groups

Let $\mathrm{\Gamma }$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\ge 2$. We introduce the notion of volume for a representation $\rho:\Gamma \rightarrow PU(m,1)$where$m \geq p$. We use this notion to generalize the Mostow–Prasad rigidity theorem. More precisely, we show that given a sequence of representations ${\rho }_n:\mathrm{\Gamma }\to PU(m,1)$ such that $\lim_{n \to \infty} $\text{Vol}$(\rho_n) =$\text{Vol}$(M)$, then there must exist a sequence of elements $g_n\in PU(m,1)$ such that the representations $g_n \circ \rho_n \circ g_n^{-1}$convergetoareduciblerepresentation$\rho_\infty$ which preserves a totally geodesic copy of ${\mathbb{H}}_{\mathbb{C}}^p$ and whose ${\mathbb{H}}_{\mathbb{C}}^p$-component is conjugated to the standard lattice embedding $i:\mathrm{\Gamma }\to PU(p,1)<PU(m,1)$. Additionally, we show that the same definitions and results can be adapted when $\mathrm{\Gamma }$ is a non-uniform lattice of $PSp(p,1)$ without torsion and for representations $\rho:\Gamma \rightarrow PSp(m,1)$,stillmantainingthehypothesis$m \geq p \geq 2$.