Asymptotics of rational representations for algebraic groups
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911889307467776 |
|---|---|
| author | Sánchez, Lander Guerrero Souza, Henrique |
| author_facet | Sánchez, Lander Guerrero Souza, Henrique |
| contents | We study the asymptotic behaviour of the cohomology of subgroups $Γ$ 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 $Γ$ 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. Lück'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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17360 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics of rational representations for algebraic groups Sánchez, Lander Guerrero Souza, Henrique Group Theory Geometric Topology Number Theory 20J06 (Primary) 20G05, 22E40, 11F41, 57K32 (Secondary) We study the asymptotic behaviour of the cohomology of subgroups $Γ$ 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 $Γ$ 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. Lück'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. |
| title | Asymptotics of rational representations for algebraic groups |
| topic | Group Theory Geometric Topology Number Theory 20J06 (Primary) 20G05, 22E40, 11F41, 57K32 (Secondary) |
| url | https://arxiv.org/abs/2405.17360 |