On a relation between certain cohomological invariants

Let G be a group, the supremum of the projective lengths of the injective ZG-modules and the supremum of the injective lengths of the projective ZG-modules. The invariants and were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203-223] in connection with the existence of complete cohomological functors. If is finite then [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203-223] and , where is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422-457]. Note that if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ, Arch. Math. 89 (1) (2007) 24-32] that if is finite then G admits a finite dimensional model for , the classifying space for proper actions.We conjecture that for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type .     

On cohomology rings of infinite groups

Let R be any ring (with 1), G a torsion free group and RG the corresponding group ring. Let be the cohomology ring associated with the RG-module M. Let H be a subgroup of finite index of G. The following is a special version of our main Theorem: Assume the profinite completion of G is torsion free. Then an element is nilpotent (under Yoneda’s product) if and only if its restriction to is nilpotent. In particular this holds for the Thompson group.There are torsion free groups for which the analogous statement is false.     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in R^N, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity.     

On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in , by which we obtain the positive and negative cones of are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.     

Extensions for finite Chevalley groups I

Let G be a connected semisimple algebraic group defined and split over the field with p elements, and k be the algebraic closure of . Assume further that G is almost simple and simply connected and let be the finite Chevalley group consisting of -rational points of G where q=p^r for a non-negative integer r. In this paper, formulas are found relating extensions between simple -modules and extensions over G (considered as an algebraic group over k). One of these formulas, which only holds for primes p?3(h−1) (where h is the Coxeter number of G), is then used to show the vanishing of self-extensions between simple -modules except for certain simple modules when r=1 and the underlying root system is of type A₁ or C_n.     

Classifying spaces with virtually cyclic stabilisers for certain infinite cyclic extensions

Let G?B?Z be an infinite cyclic extension of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain strictly descending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|≥2. We construct a model for , the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimal dimension of . Finally we determine the geometric dimension when G is a soluble Baumslag-Solitar group.     

Nilpotent orbits and some small unitary representations of indefinite orthogonal groups

For 2?m?l/2, let G be a simply connected Lie group with as Lie algebra, let be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra , and let be the universal enveloping algebra of . This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of .The roots with respect to the usual compact Cartan subalgebra are all ±e_i±e_j with 1?i<j?l. In the usual positive system of roots, the simple root e_m−e_m+1 is noncompact and the other simple roots are compact. Let be the parabolic subalgebra of for which e_m−e_m+1 contributes to and the other simple roots contribute to , let L be the analytic subgroup of G with Lie algebra , let , let be the sum of the roots contributing to , and let be the parabolic subalgebra opposite to .The members of are nilpotent members of . The group acts on with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on that carries a fully reducible representation of .For , let . Then λ_s defines a one-dimensional module . Extend this to a module by having act by 0, and define . Let be the unique irreducible quotient of . The representations under study are and , where and Π_S is the Sth derived Bernstein functor.For s>2l−2, it is known that π_s=π_s′ and that π_s′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m?s?2l−2 that π_s=π_s′ and that π_s′ is still unitary. The present paper shows that π_s′ is unitary for 0?s?m−1 even though π_s≠π_s′, and it relates the K spectrum of the representations π_s′ to the representation of on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in π_s′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on π_s′ does not make π_s′ unitary for s<0 and that the K spectrum of π_s′ in these cases is not related in the above way to the representation of on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra , 2?m?l/2.     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Perturbation from Dirichlet problem involving oscillating nonlinearities

In this paper we prove that if the potential has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), then under very mild conditions on the perturbation term g, for every k∈N there exists bk>0 such that

     

On the automorphisms of the spectral completion of the algebraic numbers field

Let be the absolute Galois group of Q and let A=C(G,C) be the Banach algebra of all continuous functions defined on G with values in C. Let be the conjugation automorphism of C and let B be the R-Banach subalgebra of A consisting of continuous functions f such that for all σ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on and let be the spectral completion of . Using a canonical isometry between and B we study the structure of the group of R-algebras automorphisms of and the structure of its subgroup of all automorphisms of which when restricted to give rise to elements of G. We introduce a topology on and prove that this last one is homeomorphic and group isomorphic to G.