Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets

Let be a uniform algebra, and let be a self-map of the spectrum of that induces a composition operator on . The object of this paper is to relate the notion of ``hyperbolic boundedness' introduced by the authors in 2004 to the essential spectrum of . It is shown that the essential spectral radius of is strictly less than if and only if the image of under some iterate of is hyperbolically bounded. The set of composition operators is partitioned into ``hyperbolic vicinities" that are clopen with respect to the essential operator norm. This partition is related to the analogous partition with respect to the uniform operator norm.

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

-continuity properties of the symmetric -stable process

Let be a domain of finite Lebesgue measure in and let be the symmetric -stable process killed upon exiting . Each element of the set of eigenvalues associated to , regarded as a function of , is right continuous. In addition, if is Lipschitz and bounded, then each is continuous in and the set of associated eigenfunctions is precompact.

     

Thin stationary sets and disjoint club sequences

We describe two opposing combinatorial properties related to adding clubs to : the existence of a thin stationary subset of and the existence of a disjoint club sequence on . A special Aronszajn tree on implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of which cannot acquire a club subset by any forcing poset which preserves and . We prove that the existence of a disjoint club sequence follows from Martin's Maximum and is equiconsistent with a Mahlo cardinal.

     

Kazhdan groups with infinite outer automorphism group

For each countable group we produce a short exact sequence where has a graphical presentation and is f.g. and satisfies property .

As a consequence we produce a group with property  such that is infinite.

Using the tools developed we are also able to produce examples of non-Hopfian and non-coHopfian groups with property .

One of our main tools is the use of random groups to achieve certain properties.

     

Rotation topological factors of minimal -actions on the Cantor set

In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

     

C*-algebras associated with interval maps

For each piecewise monotonic map of , we associate a pair of C*-algebras and and calculate their K-groups. The algebra is an AI-algebra. We characterize when and are simple. In those cases, has a unique trace, and is purely infinite with a unique KMS state. In the case that is Markov, these algebras include the Cuntz-Krieger algebras , and the associated AF-algebras . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and -transformations. For the case of interval exchange maps and of -transformations, the C*-algebra coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

Translation equivalence in free groups

Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements in a free group have the property that for every free isometric action of on an -tree the translation lengths of and on are equal.

     

A homotopy principle for maps with prescribed Thom-Boardman singularities

Let and be smooth manifolds of dimensions and ( ) respectively. Let denote an open subspace of which consists of all Boardman submanifolds of symbols with . An -regular map refers to a smooth map such that . We will prove what is called the homotopy principle for -regular maps on the existence level. Namely, a continuous section of over has an -regular map such that and are homotopic as sections.

     

Extension and separation properties of positive definite functions on locally compact groups

Continuing earlier work, we investigate two related aspects of the set of continuous positive definite functions on a locally compact group . The first one is the problem of when, for a closed subgroup of , every function in extends to some function in . The second one is the question whether elements in can be separated from by functions in which are identically one on .

     

On generalizations of Lavrentieff's theorem for Polish group actions

It is shown that for every Polish group that is not locally compact there is a continuous action of on a -complete subset of a Polish space such that cannot be extended to any superset of in . This answers a question posed by Becker and Kechris and shows that an earlier theorem of them is optimal in several aspects.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have .

In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on .

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.

     

Lattice-ordered Abelian groups and Schauder bases of unimodular fans

Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support of a fan . A unimodular fan over determines a Schauder basis of : its elements are the minimal positive free generators of the pointwise ordered group of -linear support functions. Conversely, a Schauder basis of determines a unimodular fan over : its maximal cones are the domains of linearity of the elements of . The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, is finitely generated projective iff it can be presented by a purely lattice-theoretical word.

     

The Cech filtration and monodromy in log crystalline cohomology

For a strictly semistable log scheme over a perfect field of characteristic we investigate the canonical Cech spectral sequence abutting the Hyodo-Kato (log crystalline) cohomology of and beginning with the log convergent cohomology of its various component intersections . We compare the filtration on arising from with the monodromy operator on . We also express through residue maps and study relations with singular cohomology. If lifts to a proper strictly semistable (formal) scheme over a finite totally ramified extension of , with generic fibre , we obtain results on how the simplicial structure of (as analytic space) is reflected in .

     

Projectivity and freeness over comodule algebras

Let be a Hopf algebra and an -simple right -comodule algebra. It is shown that under certain hypotheses every -Hopf module is either projective or free as an -module and is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for -simple -module algebras.

     

Partial derivatives of a generic subspace of a vector space of forms: Quotients of level algebras of arbitrary type

Given a vector space of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace of . The main result of this paper is a lower-bound (in general sharp) for the dimensions of the spaces spanned in each degree by the partial derivatives of the forms generating , in terms of the dimensions of the spaces spanned by the partial derivatives of the forms generating the original space .

Rephrasing our result in the language of commutative algebra (where this result finds its most important applications), we have: let be a type artinian level algebra with -vector , and let, for , be the -vector of the generic type level quotient of having the same socle degree . Then we supply a lower-bound (in general sharp) for the -vector . Explicitly, we will show that, for any ,

This result generalizes a recent theorem of Iarrobino (which treats the case ).

Finally, we begin to obtain, as a consequence, some structure theorems for level -vectors of type bigger than 2, which is, at this time, a very little explored topic.

     

Structure and stability of triangle-free set systems

We define the notion of stability for a monotone property of set systems. This phenomenon encompasses some classical results in combinatorics, foremost among them the Erdos-Simonovits stability theorem. A triangle is a family of three sets such that , , are each nonempty, and . We prove the following new theorem about the stability of triangle-free set systems.

Fix . For every , there exist and such that the following holds for all : if and is a triangle-free family of -sets of containing at least members, then there exists an -set which contains fewer than members of .

This is one of the first stability theorems for a nontrivial problem in extremal set theory. Indeed, the corresponding extremal result, that for every triangle-free family of -sets of has size at most , was a longstanding conjecture of Erdos (open since 1971) that was only recently settled by Mubayi and Verstraëte (2005) for all .

     

Generalized Ahlfors functions

Let be a bordered Riemann surface with genus and boundary components. Let be a smooth family of smooth Jordan curves in which all contain the point 0 in their interior. Let and let be the family of all bounded holomorphic functions on such that and for almost every . Then there exists a smooth up to the boundary holomorphic function with at most zeros on so that for every and such that for every . If, in addition, all the curves are strictly convex, then is unique among all the functions from the family .