Polynomial growth of double coset groups from groups of polynomial growth

Determine whether, for any finitely generated group G of polynomial growth and a finite subgroup H ≤ G with |H| = n, the double coset group X = H \ G / H endowed with the n-valued multiplication (H g_1 H) * (H g_2 H) = [H g_1 h g_2 H for h ∈ H] has polynomial growth of its Cayley graph.

Background

The authors recall two constructions of n-valued groups: coset groups (G,A) with A ≤ Aut(G), and double coset groups X = H \ G / H for H ≤ G with |H| = n. They prove that coset groups built from virtually nilpotent G have polynomial growth via Gromov’s theorem.

After this positive result for coset groups, they pose an analogous question for double coset groups, asking whether the polynomial growth property persists when the n-valued group comes from double cosets rather than automorphism cosets.