Abstract
New exotic invariants for measured foliations are constructed, which we call the μ \mu -classes of a pair ( F , μ ) (\mathcal {F},\mu ) . The dependence of the μ \mu -classes on the geometry of the foliation F \mathcal {F} is examined, and the dynamics of a foliation is shown to determine the μ \mu -classes in many cases. We use the μ \mu -classes to study the classifying space B Γ S L q B{\Gamma _{S{L_q}}} of foliations with a transverse invariant volume form, and we show the homotopy groups of B Γ S L q B{\Gamma _{S{L_q}}} are uncountably generated starting in degrees q + 3 q + 3 . New invariants for groups of volume preserving diffeomorphisms also arise from the μ \mu -classes; these invariants are nontrivial and related to the geometric aspects of the group action. Relations between the μ \mu -classes and the secondary classes of a foliation are exhibited.
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