Distributional Topological Complexity of groups

We study numerical invariants d(Γ) and d(Γ) of groups recently introduced in {DJ} and independently in {KW}. We compute d for finite cyclic groups Zₚ with prime p as well as for nonorientable surfaces of genus $g>3$ (for orientable surfaces it was computed in {DJ}). We prove the formula d(G H)=max (G),d (H), (G× H)\ for torsion free groups.