Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Comparative probability on von Neumann algebras

We continue here the study begun in earlier papers on implementation of comparative probability by states. Let be a von Neumann algebra on a Hilbert space and let denote the projections of . A comparative probability (CP) on (or more correctly on is a preorder on satisfying:

with for some .

If , then either or .

If , and are all in and , , then .

A state on is said to implement a on if for , . In this paper, we examine the conditions for implementability of a CP on a general von Neumann algebra (as opposed to only type I factors). A crucial tool used here, as well as in earlier results, is the interval topology generated on by . A will be termed continuous in a given topology on if the interval topology generated by is weaker than the topology induced on by the given topology. We show that uniform continuity of a comparative probability is necessary and sufficient if the von Neumann algebra has no finite direct summand. For implementation by normal states, weak continuity is sufficient and necessary if the von Neumann algebra has no finite direct summand of type I. We arrive at these results by constructing an appropriate additive measure from the CP.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

The theorem for semisimple Hopf algebras

We give an algebraic version of a result of G. I. Kac, showing that a semisimple Hopf algebra of dimension , where is a prime and , over an algebraically closed field of characteristic 0 contains a non-trivial central group-like. As an application we prove that, if , is isomorphic to a group algebra.

     

Integer-valued polynomials on Krull rings

If is a subring of a Krull ring such that is a valuation ring for every finite index , in Spec, we construct polynomials that map into the maximal possible (for a monic polynomial of fixed degree) power of , for all in Spec simultaneously. This gives a direct sum decomposition of Int, the -module of polynomials with coefficients in the quotient field of that map into , and a criterion when Int has a regular basis (one consisting of 1 polynomial of each non-negative degree).

     

On free group algebras in division rings with uncountable center

Let be a division algebra with uncountable center . If contains a noncommutative free -algebra, then also contains the -group algebra of the free group of rank 2.

     

A theorem of Briançon-Skoda type for regular local rings containing a field

Let be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if is a minimal reduction of where is -primary, then where and is the largest ideal such that . The proof uses tight closure in characteristic and reduction to characteristic for rings containing the rationals.

     

Polynomial rings over Goldie-Kerr commutative rings II

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).

     

Non-normal, standard subgroups of the Bianchi groups

Let be a subgroup of , where is a Dedekind ring, and let be the -ideal generated by , where . The subgroup is called standard iff contains the normal subgroup of generated by the -elementary matrices. It is known that, when , is standard iff is normal in . It is also known that every standard subgroup of is normal in when is an arithmetic Dedekind domain with infinitely many units. The ring of integers of an imaginary quadratic number field, , is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group has uncountably many non-normal, standard subgroups. This result is already known for related groups like .

     

Some remarks on the variation of curve length and surface area

Consider the curve , where is absolutely continuous on . Then has finite length, and if is the -neighborhood of in the uniform norm, we compare the length of the shortest path in with the length of . Our main result establishes necessary and sufficient conditions on such that the difference of these quantities is of order as . We also include a result for surfaces.

     

Degrees of unsolvability of first order decision problems for finitely presented groups

We show that for any arithmetical -degree there is a first order decision problem such that has -degree for the free 2-step nilpotent group of rank 2. This implies a conjecture of Sacerdote.

     

Parametrizing maximal compact subvarieties

For the Lie group , let be the open orbit of Lagrangian planes of signature in the generalized flag variety of Lagrangian planes in . For a suitably chosen maximal compact subgroup of and a base point we have that the orbit of is a maximal compact subvariety of . We show that for the connected component containing in the space of translates of which lie in is biholomorphic to , where denotes with the opposite complex structure.

     

Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

A reciprocity law for certain Frobenius extensions

Let be a finite Galois extension of algebraic number fields with Galois group . Assume that is a Frobenius group and is a Frobenius complement of . Let be the maximal normal nilpotent subgroup of . If is nilpotent, then every Artin L-function attached to an irreducible representation of arises from an automorphic representation over , i.e., the Langlands' reciprocity conjecture is true for such Galois extensions.

     

Basic differential forms for actions of Lie groups

A section of a Riemannian -manifold is a closed submanifold which meets each orbit orthogonally. It is shown that the algebra of -invariant differential forms on which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on which are invariant with respect to the generalized Weyl group of , under some condition.

     

Oscillatory singular integrals on and Hardy spaces

We consider boundedness properties of oscillatory singular integrals on and Hardy spaces. By constructing a phase function, we prove that boundedness may fail while boundedness holds for all . This shows that the theory and theory for such operators are fundamentally different.

     

Topologically conjugate Kleinian groups

Two Kleinian groups and are said to be topologically conjugate when there is a homeomorphism such that . It is conjectured that if two Kleinian groups and are topologically conjugate, one is a quasi-conformal deformation of the other. In this paper generalizing Minsky's result, we shall prove that this conjecture is true when is finitely generated and freely indecomposable, and the injectivity radii of all points of and are bounded below by a positive constant.