Volume invariant and maximal representations of discrete subgroups of Lie groups

Abstract

Abstract. Let Γ be a lattice in a connected semisimple Lie group G with trivial center and no compact factors. We introduce a volume invariant for representations of Γ into G, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of Γ into G.

Rigidity at infinity for lattices in rank-one Lie groups

“…For complex lattices, this result is exactly the one obtained in [BI01] or in [KK12]. The statement regarding the quaternionic case is compatible with the result obtained in [Cor92].…”

“…The prove the strong rigidity at infinity in both the complex and the quaternionic case we introduce the notion of volume for representations ρ : Γ → G m , with m ≥ p. For uniform complex lattices the definition of volume for representations ρ : Γ → P U(m, 1) is given both by [BCG99] and by [BCG07], whereas for non-uniform complex lattices we refer to [BI01] and to [KM08]. We find another interesting approach in [KK12], where the authors use the pairing between bounded cohomology and l 1 -Lipschitz homology to define the volume of a representation. However, here we give a different version of volume to adapt this notion to the non compact case, also for quaternionic lattices.…”

On the equivalence of the definitions of volume of representations

“…In this section, M is assumed to be a Riemannian manifold with finite Lipschitz simplicial volume. Gromov [18,Section 4.4] For more detail about this, see [24].…”

On deformation spaces of nonuniform hyperbolic lattices

“…In fact, in the case of hyperbolic lattices, it turns out that all definitions of the volume of a representation give the same value [15]. For further details, see [16,Section 6] (a similar proof works for any dimension) and Section 3.…”

“…Bucher-Burger-Iozzi [2] give a definition of the volume of a representation of a nonuniform hyperbolic lattice in SO(n, 1) in the language of bounded cohomology. The authors [16] give a definition of the volume of a representation for a nonuniform lattice in a semisimple Lie group through bounded cohomology, 1 -homology and simplicial volume. In fact, in the case of hyperbolic lattices, it turns out that all definitions of the volume of a representation give the same value [15].…”