Topological automorphisms of modified Sierpiński gaskets realize arbitrary finite permutation groups

The n-dimensional Sierpiński gasket X, spanned by n+1 vertices, has (n+1)! symmetries acting as the symmetric group on the vertices. The object of this note is the remarkable observation that for n≥2 every topological automorphism of X is one of these symmetries. A modification of the arguments yields that, given any finite permutation group G≤Sn+1 acting on an (n+1)-element set, there is a finite subset T⫅X such that G is the group of topological automorphisms of X\T considered as a group acting faithfully on the vertices.