Real moduli space of stable rational curves revisted

We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $overline{{mathcal M}_{0,{n+1}}}({mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use this model to find different Hopf models of the algebraic operad of Chains and homologies of $overline{{mathcal M}_{0,{n+1}}}({mathbb R})$. In particular, we show that the operad $overline{{mathcal M}_{0,{n+1}}}({mathbb R})$ is not formal. The manifolds $overline{{mathcal M}_{0,{n+1}}}({mathbb R})$ are known to be Eilenberg-MacLane spaces for the so called pure Cacti groups. As an application of the operadic constructions we prove that for each $n$ the cohomology ring $H(overline{{mathcal M}_{0,{n+1}}}({mathbb R}),{mathbb{Q}})$ is a Koszul algebra and that the manifold $overline{{mathcal M}_{0,{n+1}}}({mathbb R})$ is not formal but is a rational $K(pi,1)$ space. We give the description of the Lie algebras associated with the lower central series filtration of the pure Cacti groups.