A new construction of semi-free actions on Menger manifolds

Abstract:

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let $G$ be a compact metric zero-dimensional group, represented as the direct product of subgroups $G_{i}$, $M$ a $\mu ^{n}$-manifold and $\nu (M)$ (resp., $\Sigma (M)$) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets $X_{i}, i\ge 1,$ of $M$, there exists a $G$-action on $M$ such that (1) $\nu (M)$ and $\Sigma (M)$ are invariant subsets of $M$; and (2) each $X_{i}$ is the fixed point set of any element $g\in G_{i}\setminus \{e \}$.