Quantitative asymptotic cohomology error term

Establish that for any semi-simple algebraic group G over C and any torsion-free subgroup Γ ≤ G(C) of type FP∞, for every nonnegative integer i and each irreducible rational C-representation W of G with highest-weight parameters (λ_1,…,λ_n), the error between the normalized cohomology dimension and the i-th ℓ2-Betti number satisfies | dim H^i(Γ, W) / dim W − b_i^{(2)}(Γ) | = O(1 / min{λ_1,…,λ_n}).

Background

Beyond convergence of normalized cohomology dimensions to ℓ2-Betti numbers, the authors propose a quantitative rate of convergence governed by the smallest highest-weight parameter. This strengthens the qualitative conjecture by predicting a uniform O(1/min λ_i) error term.

Such a bound would give effective control on asymptotics, feeding into applications like growth bounds for spaces of cusp forms and computations of ℓ2-Betti numbers in geometric settings.

References

As for question (3), we conjecture the following stronger statement: Conjecture If \Gamma is torsion-free and \lambda_i = \lambda_i(k) denote the highest weight parameters of the representations W_k, then \begin{equation}\label{eq-normalized-cohomology-err} \left|\frac{\dim \HHi(\Gamma, W_k)}{\dim W_k} - b_i{(2)}(\Gamma)\right| = O\left(\frac{1}{\min {\lambda_1,\ldots,\lambda_n}\right)\,.\tag{(**)} \end{equation}

eq-normalized-cohomology-err:

$\left|\frac{\dim \HH^i(\Gamma, #1 W_k)}{\dim #1 W_k} - b_i^{(2)}(\Gamma)\right| = O\left(\frac{1}{\min \{\lambda_1,\ldots,\lambda_n\}}\right)\,.\tag{\(**\)} $

— Asymptotics of rational representations for algebraic groups (2405.17360 - Sánchez et al., 27 May 2024) in Introduction, Conjecture (label: conjecture-quantitative)