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We study the asymptotic behaviour of the cohomology of subgroups of an algebraic group G with coefficients in the various irreducible rational representations of G and raise a conjecture about it. Namely, we expect that the dimensions of these cohomology groups approximate the ²-Betti numbers of with a controlled error term. We provide positive answers when G is a product of copies of SL₂. As an application, we obtain new proofs of J. Lott's and W. L\"uck's computation of the ²-Betti numbers of hyperbolic 3-manifolds and W. Fu's upper bound on the growth of cusp forms for non totally real fields, which is sharp in the imaginary quadratic case.
Sánchez et al. (Mon,) studied this question.