On the exponent of \(G\)-spaces and isovariant extensors (Q2812513)

It is well-known that the space \(\mathrm {exp}\, X\) of all nonempty compact subsets of the metric \(G\)-space \(X\), equipped with the Vietoris topology, is homeomorphic to the Hilbert cube \(Q\) if and only if \(X\) is a nondegenerate Peano continuum (Curtis-Schori-West Theorem). In the paper under review, the author considers the problem of finding an equivariant version of the theorem. The main result of the paper states: for any nondegenerate Peano \(G\)-continuum \(\mathbb X\) with the action of the compact Lie group \(G\), if the free part \({\mathbb X}_{\mathrm {free}}\) is dense in \(\mathbb X\), then \({\mathrm {exp}}\, {\mathbb X}\) is equimorphic to the maximal equivariant Hilbert cube \(\mathbb Q\), and the converse holds if \(G\) is a compact abelian Lie group.