Qualitative asymptotic cohomology equals ℓ2-Betti numbers

Establish that for any semi-simple algebraic group G over C and any subgroup Γ ≤ G(C) of type FP∞ that is torsion-free, for every nonnegative integer i and every sequence {W_k} of irreducible rational C-representations of G whose highest-weight parameters (λ_1(k),…,λ_n(k)) all tend to infinity, the limit lim_{k→∞} dim H^i(Γ, W_k) / dim W_k equals b_i^{(2)}(Γ), the i-th ℓ2-Betti number of Γ.

Background

The paper studies the asymptotic behavior of cohomology groups H^i(Γ, W) as the coefficient representation W varies among irreducible rational representations of a semi-simple algebraic group G over C and grows along the weight lattice. The authors focus on subgroups Γ of G of type FP∞ and consider sequences of representations whose highest-weight coordinates all tend to infinity.

They conjecture that, after normalizing by dim W, the dimensions of these cohomology groups converge to the ℓ2-Betti numbers of Γ, providing a representation-theoretic analogue of Lück approximation. They verify the conjecture for products of copies of SL2(C) and apply it to hyperbolic 3-manifolds and cusp forms.