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Authors
Anna Wienhard
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This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the Kahler class, is used to define and characterize a special class of totally geodesic embeddings, called tight embeddings. In addition, special isometric actions of finitely generated groups on Hermitian symmetric spaces are studied. Results of a joint work with M. Burger and A. Iozzi about surface group representations are also discussed.
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Johnsons characterization of amenable groups states that a discrete group $Gamma$ is amenable if and only if $H_b^{n geq 1}(Gamma; V) = 0$ for all dual normed $mathbb{R}[Gamma]$-modules $V$. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective homomorphism $phi colon Gamma to K$ has amenable kernel $H$ if and only if the induced inflation map $H^bullet_b(K; V^H) to H^bullet_b(Gamma; V)$ is an isometric isomorphism for every dual normed $mathbb{R}[Gamma]$-module $V$. In addition, we obtain an analogous characterization for the (smaller) class of surjective homomorphisms $phi colon Gamma to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all normed $mathbb{R}[Gamma]$-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms $phi colon Gamma to K$, for which the restriction maps in bounded cohomology $H^bullet_b(K; V) to H^bullet_b(Gamma; phi^{-1}V)$ are isomorphisms for suitable dual normed $mathbb{R}[K]$-module $V$. We then extend the first and third results to spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.
A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.
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We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kahler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
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