Topological automorphisms of modified Sierpiński gaskets realize arbitrary finite permutation groups (Q1962088)

In this interesting note some topological properties of \(n\)-dimensional Sierpiński gaskets \(X\) are investigated. The author shows that (for \(n\geq 2\)) every topological automorphism of \(X\) is one of the \((n+1)!\) symmetries acting as the symmetric group on the vertex set of the underlying simplex. Furthermore it is shown that for a given finite subgroup of \(S_{n+1}\) there is a finite subset \(T\subseteq X\) such that \(G\) is the group of topological automorphisms of \(X\setminus T\) considered as a group acting faithfully on the vertices.