A group of paths in

We define a group structure on the set of compact ``minimal' paths in . We classify all finitely generated subgroups of this group : they are free products of free abelian groups and surface groups. Moreover, each such group occurs in . The subgroups of isomorphic to surface groups arise from certain topological -forms on the corresponding surfaces. We construct examples of such -forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using we construct a non-polygonal tiling problem in , that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group has applications to combinatorial tiling problems of the type: given a set of tiles and a region , can be tiled by translated copies of tiles in ?

     

Maximal subgroups in finite and profinite groups

We prove that if a finitely generated profinite group is not generated with positive probability by finitely many random elements, then every finite group is obtained as a quotient of an open subgroup of . The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203--220, we then
prove that a finite group has at most maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

Duality and Polynomial Testing of Tree Homomorphisms

Let be a fixed digraph. We consider the -colouring problem, i.e., the problem of deciding which digraphs admit a homomorphism to . We are interested in a characterization in terms of the absence in of certain tree-like obstructions. Specifically, we say that has tree duality if, for all digraphs , is not homomorphic to if and only if there is an oriented tree which is homomorphic to but not to . We prove that if has tree duality then the -colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the -property studied by Gutjahr, Welzl, and Woeginger.

We then focus on the case when itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads for which the -colouring problem is -complete. We contrast these with several families of oriented triads which have tree duality, or bounded treewidth duality, and hence polynomial -colouring problems. If , then no oriented triad with an -complete -colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad . We prove that none of the oriented triads with -complete -colouring problems given in the companion paper has tree duality.

     

Krull-Schmidt theorems in dimension 1

Let be a semiprime, module-finite algebra over a commutative noetherian ring of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated -modules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree -modules (called ``-lattices' in integral representation theory, and ``maximal Cohen-Macaulay modules' in the dimension-one situation in commutative algebra).

     

Package deal theorems and splitting orders in dimension 1

Let be a module-finite algebra over a commutative noetherian ring of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations , at maximal ideals of , is the family of all localizations of a finitely generated -module . When is semilocal we also determine which finitely generated modules over the -adic completion of are completions of finitely generated -modules.

If is an -order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order' of that can replace maximal orders in many situations in which maximal orders do not exist.

     

On quadratic forms of height two and a theorem of Wadsworth

Let and be anisotropic quadratic forms over a field of characteristic not . Their function fields and are said to be equivalent (over ) if and are isotropic. We consider the case where and is divisible by an -fold Pfister form. We determine those forms for which becomes isotropic over if , and provide partial results for . These results imply that if and are equivalent and , then is similar to over . This together with already known results yields that if is of height and degree or , and if , then and are equivalent iff and are isomorphic over .

     

On vanishing of characteristic numbers in Poincaré complexes

Let be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map is non-trivial and is a field. We will prove: if is a Poincaré complex oriented in -coefficient, all the characteristic numbers of in -coefficient vanish. Similarly, if and is a -Poincaré complex, then all the mod Wu numbers vanish. We will also show that the existence of a non-trivial derivation on with some suitable conditions implies vanishing of mod Wu numbers.

     

The automorphism group of a coded system

We give a general construction of coded systems with an automorphism group isomorphic to where is any preassigned group which has a ``continuous block presentation' (the isomorphism will map the shift to . Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains . We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.

     

A class of counterexamples to the Gel´fand-Kirillov conjecture

Let be a connected non-special semisimple algebraic group and let be a finite dimensional -representation such that has trivial generic stabilizer. Let . Then the semi-direct product is a counter-example to the Gel´fand-Kirillov conjecture.

     

Representations of monoids arising from finite groups of Lie type

A class of finite monoids constructed from a group of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of . The sandwich matrix of is decomposed into a product of matrices corresponding to maximal parabolic subgroups of .

     

Some New Homogeneous Einstein Metrics on Symmetric Spaces

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup of Isom acting transitively on we find all -homogeneous (non-symmetric) Einstein metrics on .

     

Powers in Finitely Generated Groups

In this paper we study the set of -powers in certain finitely generated groups . We show that, if is soluble or linear, and contains a finite index subgroup, then is nilpotent-by-finite. We also show that, if is linear and has finite index (i.e. may be covered by finitely many translations of ), then is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.

     

Relatively free invariant algebras of finite reflection groups

Let be a finite subgroup of is a field of characteristic and acting by linear substitution on a relatively free algebra of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if is a pseudo-reflection group and contains the polynomial

     

Real connective K-theory and the quaternion group

Let be the real connective K-theory spectrum. We compute and for groups whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients of the fixed points of the Tate spectrum for and . The results provide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.

     

An -minimal Boolean algebra

For every linear order we define a notion of -minimal Boolean algebra and then give a consistent example of an -minimal algebra. The Stone space of our algebra contains a point such that is an example of a countably tight, initially -compact, non-compact space. This answers a question of Dow and van Douwen.

     

Cohomology of the complement of a free divisor

We prove that if is a ``strongly quasihomogeneous" free divisor in the Stein manifold , and is its complement, then the de Rham cohomology of can be computed as the cohomology of the complex of meromorphic differential forms on with logarithmic poles along , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).

     

Hardy spaces and twisted sectors for geometric models

We study the one-to-one analytic maps that send the unit disc into a region with the property that for some complex number , . These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region that characterize their rate of growth, i.e. we prove that if and only if does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.

     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .