Rigidity at infinity for lattices in rank-one Lie groups

Abstract

Let $\Gamma$ 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 $\rho :\Gamma \to PU(m,1)$ where $m\ge p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations ${\rho }_n:\Gamma \to PU(m,1)$ 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 $g_n\circ {\rho }_n\circ g_n^{-1}$ converge to a reducible representation ${\rho }_{\infty }$ 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 $i:\Gamma \to PU(p,1)<PU(m,1)$. Additionally, we show that the same definitions and results can be adapted when $\Gamma$ is a non-uniform lattice of $PSp(p,1)$ without torsion and for representations $\rho :\Gamma \to PSp(m,1)$, still mantaining the hypothesis $m\ge p\ge 2$.

19 pages. arXiv admin note: substantial text overlap with arXiv:1706.07347

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