symplectic tori in rational elliptic surfaces let e 1 p denote the rational elliptic surface with a single multiple fiber fp of multiplicity p we construct an infinite family of homologous non isotopic symplectic tori representing the primitive 2 dimensional homology class fp in e 1 p when p 1 as a consequence we get infinitely many non isotopic symplectic tori in the fiber class of the rational elliptic surface open image in new window we also show how these tori can be non isotopically embedded as homologous symplectic submanifolds in other symplectic 4 manifolds

the lower central series and pseudo anosov dilatations the theme of this paper is that algebraic complexity implies dynamical complexity for pseudo anosov homeomorphisms of a closed surface s g of genus g penner proved that the logarithm of the minimal dilatation for a pseudo anosov homeomorphism of s g tends to zero at the rate 1 g we consider here the smallest dilatation of any pseudo anosov homeomorphism of s g acting trivially on gamma gamma k the quotient of gamma pi 1 s g by the k rm th term of its lower central series k geq 1 in contrast to penner s asymptotics we prove that this minimal dilatation is bounded above and below independently of g with bounds tending to infinity with k for example in the case of the torelli group cal i s g we prove that l cal i s g the logarithm of the minimal dilatation in cal i s g satisfies 197 l cal i s g 4 127 in contrast we find pseudo anosov mapping classes acting trivially on gamma gamma k whose asymptotic translation lengths on the complex of curves tend to 0 as g to infty

roots in the mapping class groups the purpose of this paper is the study of the roots in the mapping class groups letbe a compact oriented surface possibly with boundary let p be a finite set of punctures in the interior of and let m p denote the mapping class group of p we prove that if is of genus 0 then each f m has at most one m root for all m 1 we prove that if is of genus 1 and has non empty boundary then each f m has at most one m root up to conjugation for all m 1 we prove that however ifis of genus 2 then there exist f g m p such that f 2 g 2 f is not conjugate to g and none of the conjugates of f commutes with g afterwards we focus our study on the roots of the pseudo anosov elements we prove that if 6 then each pseudo anosov element f m p has at most one m root for all m 1 we prove that however if and the genus of is 2 then there exist two pseudo anosov elements f g m explicitely constructed such that f m g m for some m 2 f is not conjugate to g and none of the conjugates of f commutes with g furthermore if the genus ofis 0 mod4 then we can take m 2 finally we show that if is a pure subgroup of m p and f then f has at most one m root in for all m 1 note that there are finite index pure subgroups in m p

non isotopic symplectic tori in the same homology class for any pair of integers n 1 and q 2 we construct an infinite family of mutually non isotopic symplectic tori representing the homology class q f of an elliptic surface e n where f is the homology class of the fiber we also show how such families can be non isotopically and symplectically embedded into a more general class of symplectic 4 manifolds

positive configurations of flags in a building and limits of positive representations parreau compactified the hitchin component of a closed surface s of negative euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of pi 1 s on an mathbb r euclidean building in this paper we use the positivity properties of hitchin representations introduced by fock and goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

determining thurston classes using nielsen types in previous work 3 we showed how the thurston or bers classifications of diffeomorphisms of surfaces could be obtained using the nielsen types of the lifts of the diffeomorphism to the unit disc in this paper we find improved conditions on the nielsen types for the thurston and bers classes we use them to verify that an example studied by nielsen is a pseudo anosov diffeomorphism with stretching factor of degree 4 this example is of interest in its own right but it also serves to illustrate exactly how the nielsen types are used for verifying examples we discuss the general usefulness of this method

dehn twists and pseudo anosov diffeomorphisms we show that it is possible to obtain many pseudo anosov diffeomorphisms from dehn twists in particular we generalize a theorem of long and morton to obtain that iff is a pseudo anosov diffeomorphism of an oriented surface andt is the dehn twist around the simple closed curve then the isotopy class oft n f contains a pseudo anosov diffeomorphism except for at most 7 consecutive values ofn

new proofs of some results of nielsen given a compact oriented surface m with negative euler characteristic and a homeomorphism z m2 m2 what is the best representative cp m2 m2 in the isotopy class of s thurston s answered this question by defining pseudo anosov homeomorphisms a generalization of anosov diffeomorphisms on t2 which may be characterized as follows the homeomorphism rp is pseudo anosov iff there is a pair of transverse measured geodesic laminations i and au which intersect each closed nonperipheral geodesic in h4 which are topologically preserved by cp and which have their respective transverse measures multiplied under the action of cp by l n and d for some l 1 thurston showed that cp m m2 can be pieced together from homeomorphisms of subsurfaces which are either periodic or pseudoanosov more precisely thurston proved theorem 0 1 the preprint s provides an introduction to this theorem describing many of the useful properties of pseudo anosov diffeomorphisms the proof outlined in s is contained in 111

fixed points of pseudo anosov diffeomorphisms of surfaces this article supplements richard miller s paper 6 by establishing a result on the minimum number of fixed points in the homotopy class of a pseudo anosov surface transformation we thus verify one of the many farreaching results that william thurston announced in 1977 in a preprint s dealing with the classification up to homotopy of diffeomorphisms of compact surfaces thurston stated his theorems without proofs but with many hints at techniques fathi laudenbach poenaru and other workers at an intensive seminar at the university of orsay gave detailed proofs of some but not all of thurston s assertions and published their findings in a monograph 7 more recently miller 6 and gilman 2 have verified thurston s claim that his theorem 4 s p 81 follows from the classical investigations of jakob nielsen 4 using miller s exposition of the nielsenthurston theory we establish

lagrangian tori in homotopy elliptic surfaces let e 1 k denote the symplectic four manifold homotopy equivalent to the rational elliptic surface corresponding to a fibred knot k in s 3 constructed by r fintushel and r j stern in 1998 we construct a family of nullhomologous lagrangian tori in e 1 k and prove that infinitely many of these tori have complements with mutually non isomorphic fundamental groups if the alexander polynomial of k has some irreducible factor which does not divide t n 1 for any positive integer n we also show how these tori can be non isotopically embedded as nullhomologous lagrangian submanifolds in other symplectic 4 manifolds

on the nielsen type and the classification for the mapping class group let s be a compact surface of genus g with n boundary components with 2g 2 n 0 and let m s be its mapping class group also known as the teichmiiller modular group using measurd foliations thurston 15 has shown that m s contains three different types of elements bers has shown this decomposition can be obtained using teichmiiller distance 3 subdividing one of the classes further he obtains four classes nielsen has a series of papers 7 8 parts i iii in which he develops an elaborate theory of fixed points of mapping classes if t is any homeomorphism of s and h a lift of t or of a power of 1 to the upper half plane nielsen s theory involves the assignment of a pair of integers to h called the nielsen type of h the purpose of this paper is to show how the thurston classes and the bers classes can be defined in terms of the nielsen types of the lifts of

on maximal abelian subgroups in finite 2 groups we determine here up to isomorphism the structure of any finite nonabelian 2 group g in which every two distinct maximal abelian subgroups have cyclic intersection we obtain five infinite classes of such 2 groups theorem 1 1 this solves for p 2 the problem nr 521 stated by berkovich in preparation the more general problem nr 258 stated by berkovich in preparation about the structure of finite nonabelian p groups g such that a b z g for every two distinct maximal abelian subgroups a and b is treated in theorems 3 1 and 3 2 in corollary 3 3 we get a new result for an arbitrary finite 2 group as an application of theorems 3 1 and 3 2 we solve for p 2 a problem of heineken mann problem nr 169 stated in berkovich in preparation classifying finite 2 groups g such that a z g is cyclic for each maximal abelian subgroup a theorem 4 1

completely periodic directions and orbit closures of many pseudo anosov teichmueller discs in q 1 1 1 1 in this paper we investigate the closure of a large class of teichmuller discs in the stratum mathcal q 1 1 1 1 or equivalently in a rm gl 2 mathbb r invariant locus mathcal l of translation surfaces of genus three we describe a systematic way to prove that the rm gl 2 mathbb r orbit closure of a translation surface in mathcal l is the whole locus mathcal l the strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of ratner s theorem to unipotent subgroups acting on an adequate splitting this analysis applies for example to all teichmuller discs obtained by the thurston veech s construction with a trace field of degree three which are moreover obviously not veech we produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on mathcal l contrary to the situation in genus two we also study completely periodic directions on translation surfaces in mathcal l for instance we prove that completely periodic directions are dense on surfaces obtained by the thurston veech s construction

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

a weakly mixing tiling dynamical system with a smooth model we describe a weakly mixing one dimensional tiling dynamical system in which the tiling space is modeled by a surface m of genus 2 the tiling system satisfies an inflation and the inflation map is modeled by a pseudo anosov diffeomorphism d on m the expansion coefficient for d is a non pisot number in particular the leaves of the expanding foliation for d are tiled by their visits to the elements of a markov partition for d the tiling dynamical system is an almost 1 1 extension of the unit speed flow along these leaves

deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1 deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod ulus 1 the functions we consider are the corresponding eigen functions in theorem 1 1 we prove that the limit inferior of the ergodic sums n x 0 x n 1 n n is bounded for every point x in the phase space in theorem 1 2 we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length under additional assumptions we prove that ergodic integrals satisfy the central limit theorem theorem 1 3 theorem 1 9

degenerations of hyperbolic structures ii measured laminations in 3 manifolds that paper concerned the general theory of groups acting on r trees and the relationship of these actions to representations into sl2 c the purpose of the present paper is to develop the foundations of a general theory of measured laminations in 3 manifolds in the third paper 8 the results of this paper will be applied to study actions of 3 manifold groups on trees the starting point of 8 is the idea that dual to an action of qg m on an r tree is a codimension 1 measured lamination in m combining the latter results with 7 gives information about how hyperbolic structures on 3 manifolds can degenerate we view measured laminations in 3 manifolds as generalizations of both geodesic laminations on surfaces and surfaces in 3 manifolds most of this paper consists in extending thurston s theory of geodesic laminations and the hakenstallings waldhausen theory of incompressible surfaces to our context the paper is organized along the following lines in chapter i we develop the basic notions of codimension 1 measured laminations we prove one new and quite useful result theorem i 3 2 concerning a decomposition of these objects we also give a definition of the euler characteristic of a measured lamination and show that it has all the usual properties in chapter ii we introduce the branched surfaces of williams 16 and hatcher thurston we establish analogues for measured laminations carried by branched surfaces of some results of thurston s for train tracks and geodesic laminations by imitating an argument of plante s 10 we show that the leaves of a measured lamination have polynomial growth this is used to show that if a branched surface n carries some lamination all of whose leaves have virtually abelian fundamental groups and carries no disks or spheres then all laminations carried by n have zero euler characteristic we express the latter property by saying that n is flat in chapter iii generalizing haken s notion of normal surfaces we develop the theory of normal measured laminations and of normal branched surfaces in a

ideal triangle groups dented tori and numerical analysis we prove the goldman parker conjecture a complex hyperbolic ideal triangle group is discretely embedded in pu 2 1 if and only if the product of its three standard generators is not elliptic we also prove that such a group is indiscrete if the product of its three standard generators is elliptic a novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates

infinitely many hyperbolic 3 manifolds which contain no reebless foliation calegari ca has shown that if m is also atoroidal then or1 m is gromov negatively curved furthermore thurston has proposed an approach to demonstrating geometrization for such m many 3 manifolds contain reebless foliations and it has often been conjectured that all closed hyperbolic 3 manifolds do it is our impression that for many years hatcher provided the sole voice of dissent in this paper we give the first examples of closed hyperbolic 3 manifolds which contain no reebless foliation

a bayesian approach to diagnosis of asset pricing models a large literature has arisen which exploits a particular portfolio on the mean variance frontier determining a minimum variance bound on the set of stochastic discount factors state price to probability ratios this paper proposes a new variational characterization of the closely related set of state price densities based on minimization of the kullback leibler information criterion in contrast to the variance bound the resulting information bound automatically satisfies an important positivity constraint furthermore the information bound is determined by a portfolio which maximizes expected cara utility several interpretations of the information bound are given and empirical uses of it are illustrated

luttinger surgery along lagrangian tori and non isotopy for singular symplectic plane curves we discuss the properties of a certain type of dehn surgery along a lagrangian torus in a symplectic 4 manifold known as luttinger s surgery and use this construction to provide a purely topological interpretation of a non isotopy result for symplectic plane curves with cusp and node singularities due to moishezon 9

homologous non isotopic symplectic surfaces of higher genus we construct an infinite family of homologous non isotopic symplectic surfaces of any genus greater than one in a certain class of closed simply connected symplectic four manifolds our construction is the first example of this phenomenon for surfaces of genus greater than one

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics we prove that the stable holonomies of a proper codimension 1 attractor lambda for a c r diffeomorphism f of a surface are not c1 theta for theta greater than the hausdorff dimension of the stable leaves of f intersected with lambda to prove this result we show that there are no diffeomorphisms of surfaces with a proper codimension 1 attractor that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor

quantum representations and monodromies of fibered links abstract andersen masbaum and ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo anosov mapping classes to elements of infinite order for large enough level r in this paper we relate the amu conjecture to a question about the growth of the turaev viro invariants t v r of hyperbolic 3 manifolds we show that if the r growth of t v r m for a hyperbolic 3 manifold m that fibers over the circle is exponential then the monodromy of the fibration of m satisfies the amu conjecture building on earlier work 11 we give broad constructions of oriented hyperbolic fibered links of arbitrarily high genus whose s o 3 turaev viro invariants have exponential r growth as a result for any g n 2 we obtain infinite families of non conjugate pseudo anosov mapping classes acting on surfaces of genus g and n boundary components that satisfy the amu conjecture we also discuss integrality properties of the traces of quantum representations and we answer a question of chen and yang about turaev viro invariants of torus links

dual garside structure and reducibility of braids abstract benardete gutierrez and nitecki showed an important result which relates the geometrical properties of a braid as a homeomorphism of the punctured disk to its algebraic garside theoretical properties namely they showed that if a braid sends a standard curve to another standard curve then the image of this curve under the action of each factor of the left normal form of the braid with the classical garside structure is also standard we provide a new simple geometric proof of the result by benardete gutierrez nitecki which can be easily adapted to the case of the dual garside structure of braid groups with the appropriate definition of standard curves in the dual setting this yields a new algorithm for determining the nielsen thurston type of braids

the space of incompressible surfaces in a 2 bridge link complement projective lamination spaces for 2 bridge link complements are computed explicitly in this paper we construct a polyhedron p s3 lp q whose rational points correspond bijectively in a natural way with the projective isotopy classes of incompressible surfaces in the exterior of a 2 bridge link lp q c s3 here projective means that we factor out by scalar multiplication taking any number of parallel copies of a surface surfaces are not assumed to be connected we expect that p s3 lp q will turn out to be the projective lamination space of s3 lp q33 as defined and studied for general compact irreducible 3 manifolds in 5 and 12 an unexpected complication not present in thurston s theory of projective lamination spaces for surfaces is the fact that p s3 lp q is frequently noncompact for example for the whitehead link l3 8 see figure 5 4 upper left hand corner however as in the general theory p s3 lp q has a natural compactification p7 53 lp q which is a finite polyhedron to construct p s3 lp q 3 we first find a fairly natural finite collection of branched surfaces bi c s3 lp q which carry all the incompressible surfaces in s3lp q to each bi is associated a convex cell ci whose rational points parametrize the projective classes of surfaces carried by bi diserent rational points of ci can determine isotopic surfaces however due to the possibility of pushing parts of surfaces across product regions in the complement of bi the analogue of digon regions in the complement of a train track on a surface this leads to a linear projection pi ci ci f ci onto another convex polyhedral cell ci such that over the interior of ci projective isotopy classes of surfaces coincide with fibers of pi however these isotopy relations may not persist over the boundary of ci namely passing to a face of ci corresponds to passing to a branched subsurface of bi and a nonproduct complementary region of this branched subsurface may be decomposed by bi into product complementary regions of bi in this case rational points of this face of ci correspond not to projective isotopy classes of incompressible surfaces but to incompressible surfaces with these limiting phantom isotopy relations the space p7 53 lp q is foed from the cells ci by identifying their faces in the most natural way distinguishing different phantom isotopies between the same sets of surfaces pz s3 lp q consists of the open cells of ps s3 lp q for which the phantom isotopies are actual isotopies received by the editors april 12 1986 1980 mathematics subject classification 1985 revtaion primary 57m25

cantor exchange systems and renormalization we prove a one to one correspondence between i c1 conjugacy classes of c1 h cantor exchange systems that are c1 h fixed points of renormalization and ii c1 conjugacy classes of c1 h diffeomorphisms f with a codimension 1 hyperbolic attractor lambda that admit an invariant measure absolutely continuous with respect to the hausdorff measure on lambda however we prove that there is no c1 alpha cantor exchange system with bounded geometry that is a c1 alpha fixed point of renormalization with regularity alpha greater than the hausdorff dimension of its invariant cantor set

a recipe for short word pseudo anosovs given any generating set of any subgroup g of the mapping class group of a surface we find an element f with word length bounded by a constant k depending only on the surface and with the property that the minimal subsurface supporting a power of f is as large as possible for elements of g in particular if g contains a pseudo anosov map we find one of word length at most k we also find new examples of convex cocompact free subgroups of the mapping class group