Hyperspaces of Keller compacta and their orbit spaces (Q2019144)

This is a nice application of infinite-dimensional topology to topological dynamics. The authors prove among other things that for a compact group \(G\) acting affinely on a centrally symmetric Keller cube \(K\), the orbit spaces \(2^K/G\) and \(cc(K)/G\) are both homeomorphic to the Hilbert cube. Here \(2^K\) denotes the hyperspace of all non-empty closed subsets of \(K\) endowed with the Vietoris topology. Moreover, \(cc(K)\) denotes its subspace consisting of all nonempty closed and convex subsets of \(K\). The Hilbert cube \(Q\) is a centrally symmetric Keller cube, so this result applies to \(Q\). The paper ends with some interesting unsolved problems.