Let μn be the n-dimensional universal Menger compactum, X a Z-set in μn and G a metrizable zero-dimensional compact group with e the unit. It is proved that there exists a semi-free G-action on μn such that X is the fixed point set of every g ∈ G r {e}. As a corollary, it follows that each compactum with dim 6 n can be embedded in μn as the fixed point set of some semi-free G-action on μn. In [Dr], Dranishnikov showed that every metrizable zero-dimensional compact group G acts freely on the n-dimensional universal Menger compactum μ (cf. [Sa]). Here we consider the fixed point sets of semi-free actions of G on μ. A closed set X in μ is called a Z-set if there are maps f : μ → μ r X arbitrarily close to id. The following is our result: Theorem. Let G be a metrizable zero-dimensional compact group with e the unit and X a Z-set in μ. Then there exists a semi-free G-action on μ such that X is the fixed point set of every g ∈ G r {e}. By [Be, 2.3.8], each compactum X with dimX 6 n can be embedded in μ as a Z-set. Then we have the following: Corollary. Let G be a metrizable zero-dimensional compact group. Each compactum X with dimX 6 n can be embedded in μ as the fixed point set of some semi-free G-action on μ. In the proof below, for two simplicial complexes K and L, K × L denotes the simplicial complex defined as the barycentric subdivision of the cell complex { σ×τ ∣∣ σ ∈ K, τ ∈ L}. For any simplicial map f : K → L, the simplicial mapping cylinder of f is denoted by M(f) (cf. [Wh, §6]). Notice that K and L are subcomplexes of M(f). By K, we denote the set of vertices (0-skeleton) of K. Proof of Theorem. We may only consider the case that G is non-trivial, i.e., G 6= {e}. By a well-known theorem of Pontryagin [Po, §46, C], G is the inverse limit of Received by the editors April 16, 1994 and, in revised form, April 28, 1996. 1991 Mathematics Subject Classification. Primary 54F15, 54H25, 54H15; Secondary 57S10, 22C05.