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Asymptotics of rational representations for algebraic groups

Published 27 May 2024 in math.GR, math.GT, and math.NT | (2405.17360v1)

Abstract: We study the asymptotic behaviour of the cohomology of subgroups $\Gamma$ 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 $\ell2$-Betti numbers of $\Gamma$ with a controlled error term. We provide positive answers when $G$ is a product of copies of $SL_2$. As an application, we obtain new proofs of J. Lott's and W. L\"uck's computation of the $\ell2$-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.

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