The structure of some virtually free pro- groups

We prove two conjectures on pro- groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro- group which has an open free pro- subgroup of index is a free pro- product , where the are free pro- of finite rank and the are cyclic of order . The second says that if is a free pro- group of finite rank and is a finite -group of automorphisms of , then is a free factor of . The proofs use cohomology, and in particular a ``Brown theorem' for profinite groups.

     

Gaussian estimates and regularized groups

We show that if a bounded analytic semigroup on satisfies a Gaussian estimate of order and is the generator of its consistent semigroup on , then generates a -regularized group on where . We obtain the estimate of () and the -independence of , and give applications to Schrödinger operators and elliptic operators of higher order.

     

The nonquadratic imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers.

     

An extension of a convexity theorem of the generalized numerical range associated with

For any , let be the following subset of :

We show that if , then is always convex. When , it is an ellipsoid, probably degenerate. The convexity result is best possible in the sense that if we have defined similarly, then there are examples which fail to be convex when and .

The set is also symmetric about the origin for all , and contains the origin when . Equivalent statements of this result are given. The convexity result for is similar to Au-Yeung and Tsing's extension of Westwick's convexity result for .

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

     

Finite Group Schemes over Bases with Low Ramification

Let A be a complete characteristic (0,p) discrete valuation ring with absolute ramification degree e and a perfect residue field. We are interested in studying the category FF _A' of finite flat commutative group schemes over A withp-power order. When e= 1, Fontaine formulated the purely linear algebra notion of a finite Honda system over A and constructed an anti-equivalence of categories betweenineFF _A'> and the category of finite Honda systems over A when p> 2. We generalize this theory to the case e – 1.     

On the Classification of Flag-Transitive c.c*- Geometries

Let ( G) be a flag-transitive c.c*-geometry whose point-stabilizer is not an affine group. We list all known examples and show that, if (, G) is a minimal unknown example, then G is an almost simple group and the commutator subgroup G is a simple group of Lie type.     

Four-Dimensional Compact Projective Planes with a Solvable Collineation Group

We consider a four-dimensional compact projective plane whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil×R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag p W, acts transitively on and fixes no point in the set W\p. Under these conditions, we will prove that either contains a three-dimensional group of elations or acts doubly transitively on .     

Homotopy of Non-Modular Partitions and the Whitehouse Module

We present a class of subposets of the partition lattice _n with the following property: The order complex is homotopy equivalent to the order complex of _n – 1, and the S _n -module structure of the homology coincides with a recently discovered lifting of the S _n – 1-action on the homology of _n – 1. This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet P _n ⁿ – 1 of the lattice of set partitions _n , obtained by removing all elements with a unique nontrivial block. More generally, for 2 k n – 1, let Q _n ^k denote the subposet of the partition lattice _n obtained by removing all elements with a unique nontrivial block of size equal to k, and let P _n ^k = _{i = 2} ^k Q _n ⁱ . We show that P _n ^k is Cohen-Macaulay, and that P _n ^k and Q _n ^k are both homotopy equivalent to a wedge of spheres of dimension (n – 4), with Betti number . The posets Q _n ^k are neither shellable nor Cohen-Macaulay. We show that the S _n -module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property.     

Construction of Certain Cyclic Distance-Preserving Codes Having Linear-Algebraic Characteristics

An ordered list of binary words of length n is called a distance-preserving m, n-code, if the list distance between two words is equal to their Hamming distance, for distances up to m. A technique for constructing cyclic m, n-codes is presented, based on the standard Gray code and on some simple tools from linear algebra.