Asymptotic growth of the topological Poljak–Rödl function for nice groups

Determine whether, for every nice finite group—that is, every finite group that is cyclic of prime-power order or generalized quaternion—the topological Poljak–Rödl function T_G(n) satisfies \(\lim_{n\to\infty}T_G(n)=\infty\).

Background

The paper defines the topological Poljak–Rödl function T_G(n) as the minimum of $\operatorname{ind}_G(X\times Y)$ over finite free G-simplicial complexes X and Y with $\operatorname{ind}_G(X)=\operatorname{ind}_G(Y)=n$. The generalized topological Hedetniemi conjecture for G is the stronger assertion that T_G(n)=n for every n.

The main theorem proves that T_G(n)=0 for every positive n when G is not nice. The unresolved asymptotic question concerns the complementary class of nice finite groups, namely cyclic groups of prime-power order and generalized quaternion groups, and asks whether the product index nevertheless becomes arbitrarily large as the common factor index tends to infinity.