Fixed point sets of transformation groups of Menger manifolds, their pseudo-interiors and their pseudo-boundaries

Let G be a compact separable zero-dimensional group with the unit element e. We construct semifree G-actions on Menger manifolds with G-invariant pseudo-interiors and pseudo-boundaries. The main purpose of this paper is to prove the following: For each closed subset X of a Menger manifold M, there exists a semifree G-action on M such that X is the fixed point set of any g ϵ G ⧹ {e}. This gives the affirmative answers to the questions in K. Sakai, Preprint and in A. Chigogidze, K. Kawamura and E.D. Tymchatyn, Problems 6.4.3, 6.4.4 in full generality. The pseudo-interiors and the pseudo-boundaries versions of the theorem above are also given. Moreover, free G-actions on pseudo-interiors and pseudo-boundaries of Menger manifolds are constructed.