Rigidity at infinity for lattices in rank-one Lie groups

Let Gamma 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 rho : Gamma -> 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 rho(n) : Gamma -> PU(m, 1) such that lim(n ->infinity )Vol(rho(n)) = Vol(M), then there must exist a sequence of elements g(n) is an element of PU(m, 1) such that the representations g(n) circle rho(n) circle g(n)(-1) converge to a reducible representation rho(infinity) which preserves a totally geodesic copy of H(C)(p )and whose H-C(p)-component is conjugated to the standard lattice embedding i : Gamma -> 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 in PSp(p, 1) without torsion and for representations rho : Gamma -> PSp(m, 1), still maintaining the hypothesis m >= p >= 2.