Let G be one of the lamplighter groups (Formula presented.) and Sub(G) the space of all subgroups of G. We determine the perfect kernel and Cantor-Bendixson rank of Sub(G). The space of all conjugation-invariant Borel probability measures on Sub(G) is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If F is a finite group and Γ an infinite group which does not have property (T), then the conjugation-invariant probability measures on Sub(Formula presented.) supported on (Formula presented.) also form a Poulsen simplex.
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