Distributional Topological Complexity of groups

We study numerical invariants $d\text{TC}(\mathrm{\Gamma })$ and $d\text{cat}(\mathrm{\Gamma })$ of groups recently introduced in \cite{DJ} and independently in \cite{KW}. We compute $d\text{TC}$ for finite cyclic groups ${\mathbb{Z}}_p$ with prime $p$ as well as for nonorientable surfaces of genus $g>3$ (for orientable surfaces it was computed in \cite{DJ}). We prove the formula $$d\text{TC}(G\ast H)=\max \{d\text{TC}(G),d\text{TC}(H),\text{cd}(G\times H)\}$$ for torsion free groups.