On distributional topological complexity of groups and manifolds

We prove the equality $\text{dTC}(\mathrm{\Gamma })=\text{TC}(\mathrm{\Gamma })$ for distributional topological complexity of torsion free hyperbolic and of torsion free nilpotent groups. For the distributional topological complexity of lens spaces we prove the inequality $\text{dTC}(L_p^n)\le 2p-1$ and for the distributional LS-category the inequality $d\text{cat}(L_p^n)\le p-1$ which turns into equality for prime $p$ and $n>p$. We use these inequalities to bring counter-examples to the product formula for $d\text{cat}$ and $\text{dTC}$.