Operator norm localization property of relative hyperbolic group and graph of groups

In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H₁,…,Hn} has operator norm localization property if and only if each Hi, i=1,2,…,n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.     

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C^∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C^∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.     

Every finitely generated group is weakly exact

We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf G-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf G-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on ?_∞-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.     

Hochschild and ordinary cohomology rings of small categories

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH^∗(kC)→H^∗(|C|,k) and prove it is split surjective, using the factorization category of Quillen [D. Quillen, Higher algebraic K-theory I, in: Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85-147] and certain techniques from functor cohomology theory. This generalizes the well-known theorems for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, disproving a conjecture of Snashall and Solberg [N. Snashall, Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. 88 (3) (2004) 705-732].     

Cohomology rings and formality properties of nilpotent groups

We analyze k-stage formality and relate resonance with this type of formality properties. For instance, we show that, for a finitely generated nilpotent group that is k-stage formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring, truncated up to degree k+1, of a finitely generated nilpotent, k-stage formal group is generated in degree 1; this criterion is necessary and sufficient for a finitely generated, 2-step nilpotent group to be k-stage formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.     

Smoothing of bump functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^∞-smooth bump which depends locally on finitely many coordinates.     

Steenrod operations on the Chow ring of a classifying space

We use the Steenrod algebra to study the Chow ring CH^*BG of the classifying space of an algebraic group G. We describe a localization property which relates a given G to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and their fusion.     

On the Heegaard Floer homology of a surface times a circle

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S¹. We determine HF⁺(Σg×S¹,s;C) completely in the case c₁(s)=0, which for g?3 was previously unknown. We show that in this case HF^∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF⁺(Σg×S¹,s;Z) contains nontrivial 2-torsion whenever g?3 and c₁(s)=0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H₁(Σg×S¹) on HF⁺(Σg×S¹,s) when c₁(s) is nonzero.     

Φ-Harmonic functions on discrete groups and the first ℓΦ-cohomology

We study the first cohomology groups of a countable discrete group G with coefficients in a G-module ?^Φ(G), where Φ is an N-function of class Δ₂(0) ∩ ?₂(0). Developing the ideas of Puls and Martin-Valette for a finitely generated group G, we introduce the discrete Φ-Laplacian and prove a theorem on the decomposition of the space of Φ-Dirichlet finite functions into the direct sum of the spaces of Φ-harmonic functions and ?^Φ(G) (with an appropriate factorization). We prove also that if a finitely generated group G has a finitely generated infinite amenable subgroup with infinite centralizer then \(\bar H^1\) (G, ?^Φ(G)) = 0. In conclusion, we show the triviality of the first cohomology group for the wreath product of two groups one of which is nonamenable.     

Smooth hypersurfaces with a CR contraction

In this paper we show that a C^∞ real hypersurface in Cn+1 of finite D'Angelo type admitting a weakly contracting local CR automorphism is CR equivalent to a weighted homogeneous hypersurface. As an application, we show that a bounded pseudoconvex domain in Cn+1 with C^∞ boundary of finite D'Angelo type with a hyperbolic orbit accumulation point is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.     

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.     

Conway’s groupCo 3 and the Dickson invariants

In this paper, we construct a map from the classifying spaceBCo ₃ of Conway’s sporadic simple groupCo ₃ to the classifying spaceBDI(4) of the new finite loop space at the prime two DI(4) of Dwyer and Wilkerson. This map has the property that it injects the mod two cohomology ofBDI(4) (which is equal to the Dickson invariants of rank four) as a subring over which the mod two cohomology ofBCo ₃ is finitely generated as a module.     

Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms

We consider the Cauchy problem for systems of semilinear hyperbolic equations. Using the L_p→L_q type estimation for the corresponding linear parts, the existence and uniqueness of weak global solutions are investigated. We also established the behavior of solutions and their derivatives as t→+∞. Using the method of test functions developed in the works (Mitidieri and Pokhozhaev, 2001 [11], Veron and Pohozaev, 2001 [12] and Caristi, 2000 [23]) we obtain the analogue of the Fujita-Hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping. It follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense.     

Flows, fixed points and rigidity for Kleinian groups

The main application of the techniques developed in this paper is to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G ₁, G ₂ cocompact Kleinian groups, any quasiconformal map pairing a G ₁-invariant pattern to a G ₂-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas and Mj (Pattern rigidity in hyperbolic spaces: duality and pd subgroups, arxiv:math.GT/08094449, 2008) who proved rigidity for Poincare Duality subgroups. Pattern rigidity is a consequence of the study conducted in this paper of the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G ₁ and a quasiconformal conjugate h ^?1 G ₂ h of a cocompact group G ₂. We show that if the conjugacy h is not conformal then this group contains a flow, i.e. a non-trivial one parameter subgroup. Mostow rigidity is an immediate consequence.     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.     

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.     

Homotopy decomposition of a group of symplectomorphisms of S×S

We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of S²×S² when the ratio of the area of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.     

Cyclic Cohomology of étale Groupoids: The General Case

We give a general method for computing the cyclic cohomology of crossed products by étale groupoids, extending the Feigin–Tsygan–Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor, and Tsygan for the convolution algebra C _c (G) of an étale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered.     

On the cohomology of highly connected covers of finite Hopf spaces

Relying on the computation of the André-Quillen homology groups for unstable Hopf algebras, we prove that if the mod p cohomology of both the fiber and the base in an H-fibration is finitely generated as algebra over the Steenrod algebra, then so is the mod p cohomology of the total space. In particular, the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.