Rigidity at infinity for lattices in rank-one lie groups

Let Γ be a non-uniform lattice in PU(p, 1) without torsion and with p ≥ 2. By following the approach developed in [S. Francaviglia and B. Klaff, Maximal volume representations are Fuchsian, Geom. Dedicata 117 (2006) 111-124], we introduce the notion of volume for a representation ρ: Γ → PU(m, 1) where m ≥ p. We use this notion to generalize the Mostow-Prasad rigidity theorem. More precisely, we show that given a sequence of representations ρn: Γ → PU(m, 1) such that limn→∞Vol(ρn) = Vol(M), then there must exist a sequence of elements gn PU(m, 1) such that the representations gn ° ρn ° gn-1 converge to a reducible representation ρ∞ which preserves a totally geodesic copy of ℍp and whose ℍp-component is conjugated to the standard lattice embedding i: Γ → PU(p, 1) < PU(m, 1). Additionally, we show that the same definitions and results can be adapted when Γ is a non-uniform lattice in PSp(p, 1) without torsion and for representations ρ: Γ → PSp(m, 1), still maintaining the hypothesis m ≥ p ≥ 2.