Double ergodicity of the Poisson boundary and applications to bounded cohomology

We prove that the Poisson boundary of any spread out non-degenerate symmetric randomwalk on an arbitrary locally compact second countable group G is doubly $\mathcal{M}$^sep-ergodic with respect to the class $\mathcal{M}$^sep of separable coefficient Banach G-modules. The proof is direct and based on an analogous property of the bilateral Bernoulli shift in the space of increments of the random walk. As a corollary we obtain that any locally compact s-compact group G admits a measure class preserving action which is both amenable and doubly $\mathcal{M}$^sep-ergodic. This generalizes an earlier result of Burger and Monod obtained under the assumption that G is compactly generated and allows one to dispose of this assumption in numerous applications to the theory of bounded cohomology.     

Foundations of the theory of bounded cohomology

In this paper we give a new approach to the theory of bounded cohomology. The ideas of relative homological algebra, modified so that they are based on a natural seminorm in the bounded cohomology, play a central role in this approach. Moreover, a new proof is given of the vanishing theorem in the bounded cohomology of simply connected spaces, and also an analog of Leray's theorem on coverings in the theory of bounded cohomology.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 69–109, 1985.     

First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

This is the first in a series of papers exploring rigidity properties of hyperbolic actions ofZ ^k orR ^k fork ≥ 2. We show that for all known irreducible examples, the cohomology of smooth cocycles over these actions is trivial. We also obtain similar Hölder and C¹ results via a generalization of the Livshitz theorem for Anosov flows. As a consequence, there are only trivial smooth or Hölder time changes for these actions (up to an automorphism). Furthermore, small perturbations of these actions are Hölder conjugate and preserve a smooth volume.     

Second bounded cohomology group

The goal of the paper is to prove that the canonical seminorm on the second bounded cohomology group is always a norm. The corresponding question remains open in the higher dimensional cases.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 117–120, 1988.     

Polynomially bounded cohomology and discrete groups

We establish the homological foundations for studying polynomially bounded group cohomology, and show that the natural map from PH^*(G;Q) to H^*(G;Q) is an isomorphism for a certain class of groups.     

Straightening and bounded cohomology of hyperbolic groups

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hⁿ_b(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hⁿ(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hⁿ_b(G,V) ? Hⁿ(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.     

A simple spectral theory of polynomially bounded solutions and applications to differential equations

Semigroup Forum - In this paper we present a simple spectral theory of polynomially bounded functions on the half line, and then apply it to study the asymptotic behavior of solutions of fractional...     

The zero-norm subspace of bounded cohomology

Let be a closed, orientable surface of genus > 1. In this paper, non-trivial elements of the third bounded cohomology with are given constructively by using both a hyperbolic metric and a singular euclidean metric on . Furthermore, it is shown that the dimension of the subspace of consisting of zero-norm elements is the cardinality of the continuum. Received: November 26, 1996     

COMPLETELY BOUNDED COHOMOLOGY OF NON-SELFADJOINT OPERATOR ALGEBRAS

The authors prove that all n-th completely bounded cohomology groups of a nest algebra T(N) acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T(N)-bimodule containing the nest algebra. They also prove that Hcbn(A, M)(?)Hcbn(A, A) for all n≥1 and a CSL algebra A with an ultraweakly closed A-bimodule M containing A.     

The second bounded cohomology of hypo-Abelian groups

Fujiwara [K. Fujiwara, The second bounded cohomology of a group with infinitely many ends, math.GR/9505208] conjectured that the second bounded cohomology of a group is zero or infinite-dimensional as a vector space over R. However, it is known that there are some linear groups for which the second bounded cohomology is not zero but finite-dimensional. In this paper, by using the transfinitely extended derived series, we prove that Fujiwara's conjecture is true for the hypo-Abelian groups, that is, groups with no non-trivial perfect subgroups.     

Continuous selections and collectively fixed point theory with applications to generalized games

Aequationes mathematicae - Using a combination of new and old ideas we present some new collectively fixed point theorems which will then generate some new equilibrium results for generalized games.     

Cut sets and normed cohomology with applications to percolation

We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of finite presentations of groups with one end, where it gives an exponential bound on the number of minimal cut sets, and thereby shows that the critical probability for percolation on these graphs is neither zero nor one. We further show for this same class of graphs that the critical probability for the coalescence of all infinite components into a single one is neither zero nor one.

     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .