Asymptotics of rational representations for algebraic groups

TL;DR

This paper investigates the asymptotic behavior of cohomology in algebraic groups, proposing a conjecture linking it to $\, ext{ extlbrackdbl} ext{-Betti} numbers, and confirms it for certain groups, with applications to hyperbolic 3-manifolds and cusp form growth.

Contribution

It introduces a conjecture relating cohomology dimensions to $\, extlbrackdbl} ext{-Betti} numbers and proves it for products of $SL_2$, providing new proofs and bounds in geometric and number theory contexts.

Findings

Confirmed the conjecture for products of $SL_2$ groups.

Provided new proofs for $\, extlbrackdbl} ext{-Betti} numbers of hyperbolic 3-manifolds.

Established sharp bounds on cusp form growth for non-totally real fields.

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 $\ell^2$-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.