Lie–Lück conjecture: qualitative rank-function convergence

Establish that for any semi-simple algebraic group G over C, any finitely generated subgroup Γ ≤ G(C), and any central character χ of Z(G) whose restriction to Z(G) ∩ Γ is fixed, as the highest-weight parameters of irreducible representations W with central character χ tend to infinity, the Sylvester matrix rank functions induced by W on C[Γ] converge to the twisted von Neumann rank function rk_Γ^χ on C[Γ].

Background

To strengthen and generalize the cohomological conjectures, the authors recast them in terms of Sylvester matrix rank functions associated to representations and von Neumann algebras. This formulation dispenses with FP∞ assumptions and works for finitely generated subgroups, potentially with torsion, by incorporating a fixed central character χ.

The conjecture asserts that the representation-induced rank function rk_W^Γ on C[Γ] converges to the twisted von Neumann rank rk_Γ^χ as all highest-weight coordinates go to infinity along the χ-isotypic subset of the weight lattice. This qualitative rank convergence implies the cohomological Conjecture on normalized cohomology dimensions.