Inequality between sequential topological complexity and cohomological dimension for epimorphisms of geometrically finite groups

Determine whether, for every epimorphism φ: Γ → Λ of geometrically finite groups and integer r≥2, the inequality cdp(φ^{r−1}) ≤ TC^r(φ) ≤ cdp(φ^r) holds, where φ^k denotes the k-fold product homomorphism φ × ⋯ × φ: Γ^k → Λ^k and TC^r(φ) denotes the sequential topological complexity of the classifying map Bφ: BΓ → BΛ.

Background

The authors develop the sequential topological complexity TC^r of maps and relate it to LS-category bounds and cohomological dimensions. For spaces, a known inequality cdp(Γ^{r−1}) ≤ TC^r(Γ) ≤ cdp(Γ^r) motivates seeking an analogous relation for homomorphisms.

Confirming this inequality for epimorphisms of geometrically finite groups would connect TC^r(φ) tightly to product cohomological dimensions of φ, refining lower and upper bounds and potentially guiding computations for broad classes of group homomorphisms.