Discrete subgroups of PU(2,1) with large critical exponents

Let Γ be a discrete subgroup of PU(2,1), the isometry group of the complex-hyperbolic space CH² of (complex) dimension two. Under a suitable normalization of the Riemannian metric of CH², the critical exponent δ(Γ) of Γ, when Γ is a lattice, is 4. In this talk, we discuss an example of a sequence (Γₙ) of discrete subgroups of PU(2,1) such that, for all n∈N, δ(Γₙ) < 4, but δ(Γₙ) → 4, as n → ∞. This example shows that Corlette’s gap theorem on critical exponents of discrete isometry groups of the quaternionic-hyperbolic spaces does not hold for CH². This talk is based on joint work with Beibei Liu.