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Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X isa Stone dual of a homogeneous Boolean algebra. It is shown that the universalminimal space M(G) of the topological group G=Homeo(X), is the space of maximalchains on X introduced by Uspenskij. If X is metrizable then clearly X ishomeomorphic to the Cantor set and the result was already known (Glasner-Weiss2003). However many new examples arise for non-metrizable spaces. Theseinclude, among others, the generalized Cantor sets X={0,1}^{kappa} fornon-countable cardinals kappa, and the corona or remainder of omega,X=beta(omega) \ omega, where beta(omega) denotes the Stone-Cechcompactification of the natural numbers.

The universal minimal space for groups of homeomorphisms of h-homogeneous spaces