Automorphisms of fiber surfaces of genus , inducing the identity in cohomology

Let be a complex non-singular projective surface of general type with a genus fibration and . Let be a non-trivial subgroup of automorphisms of , inducing trivial actions on for all . Then , and . Examples of such surfaces are given.

     

Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms on algebras and their modules over the algebraic closure of the finite field of elements, we establish a relation between the representation theory of over and that of the -fixed point algebra over . More precisely, we prove that the category    mod- of finite-dimensional -modules is equivalent to the subcategory of finite-dimensional -stable -modules, and, when is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the -orbits of the isoclasses of indecomposable -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over can be interpreted as -stable representations of the corresponding quiver over . We further prove that every finite-dimensional hereditary algebra over is Morita equivalent to some , where is the path algebra of a quiver over and is induced from a certain automorphism of . A close relation between the Auslander-Reiten theories for and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by ``folding" the Auslander-Reiten quiver of . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

     

Dynamical forcing of circular groups

In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties.

As an application, we show that the set consisting of rotation numbers which can be forced by finitely presented groups is an infinitely generated -module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number is forced by a pair , where is a finitely presented group and is some element, if the set of rotation numbers of as varies over is precisely the set .

We show that the set of subsets of which are of the form

where varies over countable groups, are exactly the set of closed subsets which contain and are invariant under . Moreover, we show that every such subset can be approximated from above by for finitely presented .

As another application, we construct a finitely generated group which acts faithfully on the circle, but which does not admit any faithful action, thus answering in the negative a question of John Franks.

     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

Prime geodesic theorem for higher-dimensional hyperbolic manifold

For a -dimensional hyperbolic manifold , we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group of to be a discrete subgroup of with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl's law, we obtained a lower estimate as goes to .

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

Length, multiplicity, and multiplier ideals

Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

     

-manifolds with planar presentations and the width of satellite knots

We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of is a motivating example. To we associate a connectivity graph . For , is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of is a tree, then there is a level-preserving reimbedding of so that is a connected sum of handlebodies.

Corollary.

The width of a satellite knot is no less than the width of its pattern knot and so

.

     

Algebraic Goodwillie calculus and a cotriple model for the remainder

Goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the Taylor series of a function. His order approximation at a space depends on the values of on coproducts of large suspensions of the space: .

We define an ``algebraic' version of the Goodwillie tower, , that depends only on the behavior of on coproducts of . When is a functor to connected spaces or grouplike -spaces, the functor is the base of a fibration

whose fiber is the simplicial space associated to a cotriple built from the cross effect of the functor . In a range in which commutes with realizations (for instance, when is the identity functor of spaces), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor in many interesting cases.

     

Non-isotopic symplectic surfaces in product 4-manifolds

Let be a closed Riemann surface of genus . Generalizing Ivan Smith's construction, we give the first examples of an infinite family of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside the product symplectic -manifolds , where and .

     

Weierstrass functions with random phases

Consider the function

where 1$">, , and is a non-constant 1-periodic Lipschitz function. The phases are chosen independently with respect to the uniform probability measure on . We prove that with probability one, we can choose a sequence of scales such that for every interval of length , the oscillation of satisfies . Moreover, the inequality is almost surely true at every scale. When is a transcendental number, these results can be improved: the minoration is true for every choice of the phases and at every scale.

     

Quivers with relations arising from clusters case)

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let be a cluster algebra of type . We associate to each cluster of an abelian category such that the indecomposable objects of are in natural correspondence with the cluster variables of which are not in . We give an algebraic realization and a geometric realization of . Then, we generalize the ``denominator theorem' of Fomin and Zelevinsky to any cluster.

     

Constant mean curvature surfaces in

The subject of this paper is properly embedded surfaces in Riemannian three manifolds of the form , where is a complete Riemannian surface. When , we are in the classical domain of surfaces in . In general, we will make some assumptions about in order to prove stronger results, or to show the effects of curvature bounds in on the behavior of surfaces in .

     

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Let be a bigraded ideal in the bigraded polynomial ring . Assume that has codimension 2. Then is a finite set of points. We prove that if is a local complete intersection, then any syzygy of the vanishing at , and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

On decompositions in homotopy theory

We first describe Krull-Schmidt theorems decomposing spaces and simply-connected co- spaces into atomic factors in the category of pointed nilpotent -complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic spaces and homotopy types of atomic co- spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed.

     

Topological obstructions to certain Lie group actions on manifolds

Given a smooth closed -manifold , this article studies the extent to which certain numbers of the form are determined by the fixed-point set , where classifies the universal cover of , , is a polynomial in the Pontrjagin classes of , and is in the subalgebra of generated by . When , various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free -action extends to an action by some semisimple compact Lie group , then . Similar vanishing results are obtained for spin manifolds admitting certain -actions.

     

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If is a complex hyperplane arrangement, with complement , we show that the Chen ranks of are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring , viewed as a module over the exterior algebra on :

   for

where is a field of characteristic 0. The Chen ranks conjecture asserts that, for sufficiently large, , where is the number of -dimensional components of the projective resonance variety . Our earlier work on the resolution of over and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of and a localization argument, we establish the inequality

   for

for arbitrary . Finally, we show that there is a polynomial of degree equal to the dimension of , such that , for all .

     

The automorphism tower of groups acting on rooted trees

The group of isometries of a rooted -ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in . This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group studied by R. Grigorchuk, and the group studied by N. Gupta and the second author.

In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as and . We describe this tower for all subgroups of which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of and .

More precisely, the tower of is infinite countable, and the terms of the tower are -groups. Quotients of successive terms are infinite elementary abelian -groups.

In contrast, the tower of has length , and its terms are -groups. We show that is an elementary abelian -group of countably infinite rank, while .