  We study numerical invariants $d\TC(\Gamma)$ and $d\cat(\Gamma)$ of groups
recently introduced in \cite{DJ} and independently in \cite{KW}. We compute
$d\TC$ 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 $$d\TC(G\ast H)=\max\{d\TC (G),d\TC (H),
\cd(G\times H)\}$$ for torsion free groups.
