A characterization of the free actions of zero-dimensional compact groups on $k$-dimensional Menger compacta

This paper contains a theorem characterizing free actions of a zerodimensional compact group $G$ on a $k$-dimensional Menger compactum $\mu^k$: two free actions $\alpha\colon G\times\mu^k\to\mu^k$ and $\alpha_1\colon G\times\mu^k\to\mu^k$ are equivalent provided that the dimensions of the orbit spaces are equal to $k$ and the actions are strongly universal with respect to the class of free compacta $Y$ with $\dim(Y/G)\leqslant k$. This theorem, as well as other results of the paper, suggest that, in the category of compact spaces equipped with free actions of groups of the above type, there are distinguished objects (referred to in what follows as free Menger compacta $\mu^k_f$), with properties analogous to those of the classical Menger compacta $\mu^k$.