Diffeomorphism groups --from a view point of rigidity problem

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<Kanai> He made computation of Gelfand-Fuks cohomology of diffeomorphism groups and their homogeneous spaces especially keeping in his mind possible applications of it to rigidity problems. A new perspective on infinitesimal rigidity of Anosov actions of higher-rank abelian groups that arise from semisimple Lie groups of rank greater than one has also been obtained by him.<Izeki> It is known that the domain of discontinuity of a convex-cocompact Kleinian group is compact. Conversely, he proved that a Kleinian group is convex cocompact provided the Hausdorff dimension of the limit set is less than n/2.<Izeki and Nayatani> They introduced a combinatorial notion of harmonic map of a simplicial complex into singular space of nonpositive curvature, and showed an existence theorem under an appropriate assumption. A fixed point theorem for an isometric action of a discrete group on a nonpositively curved space has been established as an application.<Kotani> She investigated, in a joint work of T.Sunada, the large deviation principle. Another achievement by her is the Lipschitz continuity of the bounds of the spectrum of some self-adjoint operator with magnetic, effect.<Tsuboi> A projective Anosov flow is said to be regular if its stable and unstable plane fields are integrated by smooth foliations. He proved that for a regular projective Anosov flow on a Seifert fibered space if the associated foliations have no compact leaf then it is a regular Anosov flow, and in consequence is quasi-Fuksian, due to a theorem of Ghys.<Fujiwara> In the joint work with Bestvina, he made a computation of the second bounded cohomology of the mapping class groups. It follows that a discrete subgroup of a Lie group can never be realized as a subgroup of the mapping class groups.