asymptotic dimension and small cancellation for hierarchically hyperbolic spaces and groups we prove that all hierarchically hyperbolic groups have finite asymptotic dimension one application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface improving the bound from exponential to at most quadratic in the complexity of the surface we also apply the main result to various other hierarchically hyperbolic groups and spaces we also prove a small cancellation result namely if g is a hierarchically hyperbolic group h g is a suitable hyperbolically embedded subgroup and n h is sufficiently deep in h then g n is a relatively hierarchically hyperbolic group this new class provides many new examples to which our asymptotic dimension bounds apply along the way we prove new results about the structure of hhss for example the associated hyperbolic spaces are always obtained up to quasi isometry by coning off canonical coarse product regions in the original space generalizing a relation established by masur and minsky between the complex of curves of a surface and teichmuller space

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

a probabilistic approach to rank complex fuzzy numbers abstract techniques for ranking simple fuzzy numbers are abundant in the literature however we lack efficient methods for comparing complex fuzzy numbers that are induced by arithmetic operations in this paper a probabilistic approach is taken membership functions of fuzzy numbers are first converted into probability density functions the mellin transform is then introduced to compute the mean and the variance of a complex fuzzy number the fuzzy number with the higher mean is ranked higher if the means are equal the one with the smaller variance is judged higher rank two numerical examples and a fuzzy multiple attribute decision analysis are illustrated

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

the geometry of purely loxodromic subgroups of right angled artin groups we prove that finitely generated purely loxodromic subgroups of a right angled artin group a fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups mod s in particular such subgroups are quasiconvex in a in addition we identify a milder condition for a finitely generated subgroup of a that guarantees it is free undistorted and retains finite generation when intersected with a for subgraphs of these results have applications to both the study of convex cocompactness in mod s and the way in which certain groups can embed in right angled artin groups

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

limit sets of teichm uller geodesics with minimal nonuniquely ergodic vertical foliation ii given a sequence of curves on a surface we provide con ditions which ensure that 1 the sequence is an innite quasi geodesic in the curve complex 2 the limit in the gromov boundary is repre sented by a nonuniquely ergodic ending lamination and 3 the sequence divides into a nite set of subsequences each of which projectively con verges to one of the ergodic measures on the ending lamination the conditions are suciently robust allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension 3g 3 for example we also study the limit sets in the thurston boundary of teichmuller geodesic rays dened by quadratic dierentials whose vertical foliations are obtained from the constructions mentioned above we prove that such examples exist for which the limit is a cycle the 1 skeleton of the simplex of projective classes of measures visiting every vertex

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

contracting orbits in outer space we show that strongly contracting geodesics in outer space project to parameterized quasigeodesics in the free factor complex this result provides a converse to a theorem of bestvina feighn and is used to give conditions for when a subgroup of mathrm out mathbb f has a quasi isometric orbit map into the free factor complex it also allows one to construct many new examples of strongly contracting geodesics in outer space

modeling uncertainty in multi criteria decision analysis abstract his paper provides a review of multiple criteria decision analysis mcda for cases where attribute evaluations are uncertain the main aim is to identify different tools which can be used to represent uncertain evaluations and to broadly survey the available decision models that can be used to support uncertain decision making the review includes models using probabilities or probability like quantities explicit risk measures such as quantiles and variances fuzzy numbers and scenarios the practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed

non realizability and ending laminations proof of the density conjecture we give a complete proof of the bers sullivan thurston density conjecture in the light of the ending lamination theorem it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds the ending invariants are either riemann surfaces or filling laminations supporting masur domain measured laminations and satisfying some mild additional conditions with any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence we derive the necessary compactness theorem combining the rips machine with non existence results for certain small actions on real trees of free products of surface groups and free groups we prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface in order to be able to apply the ending lamination theorem we have to prove that this algebraic limit has the desired topological type and that these non realized laminations are ending laminations that this is the case is the main novel technical result of this paper loosely speaking we show that a filling masur domain lamination which is not realized is an ending lamination

are large distance heegaard splittings generic in a previous paper we introduced a notion of genericity for countable sets of curves in the curve complex of a surface s based on the lebesgue measure on the space of projective measured laminations in s with this definition we prove that for each fixed g 1 the set of irreducible genus g heegaard splittings of high distance is generic in the set of all irreducible heegaard splittings our definition of genericity is different and more intrinsic then the one given via random walks

curve complexes are rigid any quasi isometry of the curve complex is bounded distance from a simplicial automorphism as a consequence the quasi isometry type of the curve complex determines the homeomorphism type of the surface

a new method for tool steel materials selection under fuzzy environment abstract this paper presents a new method to solve the tool steel materials selection problem under fuzzy environment where the importance weights of different criteria and the ratings of various alternatives under different criteria are assessed in linguistic terms represented by fuzzy numbers because the proposed method uses simple arithmetic operations rather than the complicated arithmetic operations for aggregation and ranking of fuzzy numbers described in wang and chang 1995 its execution is much faster than the one presented in wang and chang 1995

comparison of fuzzy values on a continuous domain abstract a comparison measure called the satisfaction function sf is extended in this paper the sf estimates the degree to which arithmetic comparisons between two fuzzy values are satisfied the previously proposed sf was defined on a discrete domain sfd so in order to compare continuous fuzzy values by the sfd the fuzzy values should be converted into discrete ones in order to directly compare continuous fuzzy values the sf is extended to a continuous domain sfc the sfc is defined based on the sfd and its properties are also developed since the sfc is extended from the sfd its properties includes those of the sfd and it has the same interpretation as the sfd thus it can be used in any applications instead of the sfd the sfc requires less computing time than the sfd under some conditions for comparisons between a fuzzy and a crisp value and between two crisp values the sfc are also extended

ranking fuzzy numbers with preference weighting function expectations based on the recent research developments of fuzzy number ranking the paper extends the centroid expectation approach and proposes a preference weighting function expectation method to rank fuzzy numbers the weighting function gives different emphasis on the possible values of the fuzzy number which can be used to represent the decision maker s pessimistic neutral or optimistic attitude an optimistic measure to represent this preference information is proposed some properties of the weighting function and optimistic measure are systematically analyzed then the paper proposes a special class weighting function with maximum entropy principle and analyzes some of its properties with this parameterized maximum entropy weighting function we can get preference expectation of any fuzzy number with given optimistic level next the weighting function is extended to represent the confidential attitude of the decision maker by giving different emphasis on the cut level set of the fuzzy number some results about confidential attitude that parallel to the valuation attitude are attained we especially extended the badd basic defuzzification distribution approach with preference expectation of fuzzy numbers and another confidential measure with badd operator is proposed finally the expected valuations of fuzzy numbers are used to ranking the fuzzy numbers compared with other fuzzy number ranking methods the method of this paper is easily interpreted robust flexible and invariant with translating and scaling for real numbers

connectivity of the space of ending laminations we prove that for any closed surface of genus at least four and any punctured surface of genus at least two the space of ending laminations is connected a theorem of e klarreich 28 theorem 1 3 implies that this space is homeomorphic to the gromov boundary of the complex of curves it follows that the boundary of the complex of curves is connected in these cases answering the conjecture of p storm other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate kleinian groups

measure equivalence rigidity of the mapping class group we show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence if the mapping class group is measure equivalent to a discrete group then they are commensurable up to finite kernels moreover we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group we obtain similar results for finite direct products of mapping class groups

schwarzian derivatives projective structures and the weil petersson gradient flow for renormalized volume to a complex projective structure on a surface thurston associates a locally convex pleated surface we derive bounds on the geometry of both in terms of the norms and 2 of the quadratic differential of given by the schwarzian derivative of the associated locally univalent map we show that these give a unifying approach that generalizes a number of important well known results for convex cocompact hyperbolic structures on 3 manifolds including bounds on the lipschitz constant for the nearest point retraction and the length of the bending lamination we then use these bounds to begin a study of the weil petersson gradient flow of renormalized volume on the space cc n of convex cocompact hyperbolic structures on a compact manifold n with incompressible boundary leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of dn the double of n

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

contracting elements and random walks we define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups pseudo anosovs in mapping class groups rank one isometries in groups acting properly on proper cat 0 spaces elements acting hyperbolically on the bass serre tree in graph manifold groups we also define a related notion of weakly contracting element and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in out fn n 3 we prove that any simple random walk in a non elementary finitely generated subgroup containing a weakly contracting element ends up in a non weakly contracting element with exponentially decaying probability also we show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup

geometry of the mapping class groups i boundary amenability we construct a geometric model for the mapping class group mathcal m mathcal c mathcal g of a non exceptional oriented surface s of genus g with k punctures and use it to show that the action of mathcal m mathcal c mathcal g on the compact metrizable hausdorff space of complete geodesic laminations for s is topologically amenable as a consequence the novikov higher signature conjecture holds for every subgroup of mathcal m mathcal c mathcal g

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

thick metric spaces relative hyperbolicity and quasi isometric rigidity we study the geometry of non relatively hyperbolic groups generalizing a result of schwartz any quasi isometric image of a non relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup this implies that a group being relatively hyperbolic with non relatively hyperbolic peripheral subgroups is a quasi isometry invariant as an application artin groups are relatively hyperbolic if and only if freely decomposable we also introduce a new quasi isometry invariant of metric spaces called metrically thick which is sufficient for a metric space to be non hyperbolic relative to any non trivial collection of subsets thick finitely generated groups include mapping class groups of most surfaces outer automorphism groups of most free groups certain artin groups and others non uniform lattices in higher rank semisimple lie groups are thick and hence non relatively hyperbolic in contrast with rank one which provided the motivating examples of relatively hyperbolic groups mapping class groups are the first examples of non relatively hyperbolic groups having cut points in any asymptotic cone resolving several questions of drutu and sapir about the structure of relatively hyperbolic groups outside of group theory teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the weil peterson metric in contrast with brock farb s hyperbolicity result in low complexity

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

positive solutions for a system of second order nonlinear multi point eigenvalue problems by using the guo krasnosel skii fixed point theorem we determine intervals for two eigenvalues for which there exist positive solutions of a system of second order nonlinear ordinary differential equations subject to multi point boundary conditions the nonexistence of positive solutions is also investigated

the hyperbolicity of the sphere complex via surgery paths handel and mosher have proved that the free splitting complex fs for the free group is gromov hyperbolic this is a deep and much sought after result since it establishes fs as a good analogue of the curve complex for surfaces we give a shorter alternative proof of this theorem using surgery paths in hatcher s sphere complex another model for the free splitting complex instead of handel and mosher s fold paths as a byproduct we get that surgery paths are unparameterized quasi geodesics in the sphere complex

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

fuzzy nash equilibriums in crisp and fuzzy games in this paper we introduce fuzzy nash equilibrium to determine a graded representation of nash equilibriums in crisp and fuzzy games this interpretation shows the distribution of equilibriums in the matrix form of a game and handles uncertainties in payoffs in addition a new method to rank fuzzy values with the user s viewpoint is investigated by this means the definition of satisfaction function which provides the result of comparison in the form of real value is developed when users have preferences regarding the payoffs

fuzzy preference and orlovsky choice procedure abstract this paper provides an axiomatic characterization of the orlovsky choice procedure for a decision maker whose preference are fuzzy in contrast to the result in banerjee 1993 our characterization does not require that in a pairwise choice the choice coincides with the orlovsky set moreover our characterization result holds for a larger domain of preferences than that in banerjee 1993

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

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