Asymptotics of rational representations for algebraic groups
| Autor: | Sánchez, Lander Guerrero, Souza, Henrique |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\ell^2$-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 $\ell^2$-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. Comment: 33 pages, comments are welcome! |
| Databáze: | arXiv |
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