No Arabic abstract
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.
We give a new perspective on the homological characterisations of amenability given by Johnson in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group.
Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of the representation theory of finite groups and thus describes a geometric representation theory. We will prove that B_G consists only of copies of Z and Z/2Z. Furthermore we will show that there is a surjection from the G-equivariant cobordism group of surface diffeomorphisms to B_G. We will define a G-signature which is related to the G-signature of Atiyah and Singer and prove that this new G-signature is injective on the copies of Z in B_G.
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.
We discuss an approach to the emph{covering} and emph{vanishing} theorems for the comparison map from bounded cohomology to singular cohomology, based on the observation that the comparison map is the coassembly map for bounded cohomology.
We show that if an inclusion of finite groups H < G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.