Lie–Lück conjecture: quantitative rank-function convergence

Establish that for any semi-simple algebraic group G over C, any finitely generated subgroup Γ ≤ G(C), any central character χ of Z(G) whose restriction to Z(G) ∩ Γ is fixed, any dominant highest weight λ ∈ X(T)_χ, and any matrix A over C[Γ], the difference between the Sylvester rank rk_W^Γ(A) induced by the irreducible representation W of highest weight λ and the twisted von Neumann rank rk_Γ^χ(A) satisfies | rk_W^Γ(A) − rk_Γ^χ(A) | = O(1 / min{λ_1,…,λ_n}).

Background

This quantitative rank-function conjecture strengthens the qualitative Lie–Lück conjecture by asserting an explicit O(1/min λ_i) error bound for every matrix over C[Γ], uniformly over weights with fixed central character χ. It provides a matrix-level analogue of the desired cohomological rate of convergence.

The authors show that this rank-function formulation implies the earlier cohomological conjectures on normalized Betti numbers and their convergence rates, and they verify the conjectures for products of groups of type A1 (i.e., products of SL2) via p-adic analytic and enveloping algebra techniques.