Stability of hybrid rapid decay under direct products of pairs

Determine whether, for any two pairs of finitely generated groups (G,H) and (Γ,Λ) that both have the hybrid rapid decay property, the product pair (G×Γ, H×Λ) also has the hybrid rapid decay property without imposing additional growth assumptions on subgroups or quotients.

Background

The paper proves that (G×Γ, H×Λ) has the hybrid rapid decay property under extra conditions (e.g., polynomial growth of Γ/Λ or Λ). The authors then raise the question of whether such stability under direct products holds in full generality.

A positive resolution would broaden the class of constructions preserving the hybrid rapid decay property and unify the currently separate sufficient conditions.

References

Even for direct product, we don't know if the fact that (G,H) has rapid decay and (Γ,Λ) has rapid decay implies that the pair (G× Γ, H× Λ) has the rapid decay property.

— The rapid decay property for pairs of discrete groups (2412.07994 - Chatterji et al., 11 Dec 2024) in Remark following Proposition (case of direct product), Section: Stability results and open questions