metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

curvature and rank of teichmuller space let 5 be a surface with genus g and n boundary components and let d s 3g 3 n denote the number of curves in any pants decomposition of s we employ metric properties of the graph of pants decompositions p s to prove that the weil petersson metric on teichmiiller space teich s is gromov hyperbolic if and only if d s 3 the weil petersson metric has higher rank in the sense of gromov it admits a quasi isometric embedding of rk k 2 when d s 3 and that no complete riemannian metric of pinched negative curvature exists on moduli space m s when d s 2 1 introduction the weil petersson metric on teichmiiller space teich 5 has many curious properties it is a riemannian metric with negative sectional curvature but its curvatures are not bounded away from zero or negative infinity it is geodesically convex but it is not complete in this paper we show that in spite of exhibiting negative curvature behavior the weil petersson metric is not coarsely negatively curved except for topologically simple surfaces s our main theorem answers a question of bowditch be question 11 4

an algebraic approach to invariant preserving integators the case of quadratic and hamiltonian invariants in this article conditions for the preservation of quadratic and hamiltonian invariants by numerical methods which can be written as b series are derived in a purely algebraical way the existence of a modified invariant is also investigated and turns out to be equivalent up to a conjugation to the preservation of the exact invariant a striking corollary is that a symplectic method is formally conjugate to a method that preserves the hamitonian exactly another surprising consequence is that the underlying one step method of a symmetric multistep scheme is formally conjugate to a symplectic p series when applied to newton s equations of motion

order conditions for canonical runge kutta schemes when numerically integrating hamiltonian systems of differential equations it is often advantageous to use canonical methods i e methods that preserve the symplectic structure of the phase space thus reproducing an important feature of the hamiltonian flow an s stage runge kutta rk method without redundant stages is canonical if and only if with a standard notation b i a ij b j a ji b i b j 0 1 leqq i j leqq s it is shown that for canonical rk methods there are many redundancies in the standard order conditions for a canonical method to have order p it is sufficient that the b i s a ij s satisfy a system of algebraicequations that has roughly speaking an equation per nonrooted tree of order leqq p furthermore a new methodology is presented for the investigation of the order of canonical integration methods not necessarily rk methods when applied to hamiltonian systems in the new approach consistency is studied by taylor expanding a suitable scalar function in ter

a measure zero universal differentiability set in the heisenberg group we show that the heisenberg group mathbb h n contains a measure zero set n such that every lipschitz function f mathbb h n rightarrow mathbb r is pansu differentiable at a point of n the proof adapts the construction of small universal differentiability sets in the euclidean setting we find a point of n and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal then deduce pansu differentiability at such a point

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

towards the resolution of the gibbs phenomena it is well known that the expansion of an analytic nonperiodic function on a finite interval in a fourier series leads to spurious oscillations at the interval boundaries this result is known as the gibbs phenomenon the present paper introduces a new method for the resolution of the gibbs phenomenon which follows on the reconstruction method of gottlieb and coworkers siam rev 39 1997 644 based on gegenbauer polynomials orthogonal with respect to weight function 1 x2 1 2 we refer to their approach as the direct method and to the new methodology as the inverse method both methods use the finite set of fourier coefficients of some given function as input data in the re expansion of the function in gegenbauer polynomials or in other orthogonal basis sets the finite partial sum of the new expansion provides a spectrally accurate approximation to the function in the direct method this requires that certain conditions are met concerning the parameter in the weight function the number of fourier coefficients n and the number of gegenbauer polynomials m we show that the new inverse method can give exact results for polynomials independent of and with m n the paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method

almost bi lipschitz embeddings using covers of balls centred at the origin abstract in olson and robinson 2010 8 introduced the notion of an almost homogeneous metric space and showed that if x is a subset of a hilbert space such that x x is almost homogeneous then x admits almost bi lipschitz embeddings into euclidean spaces in this paper we extend this result and we show that if x is a subset of a banach space such that x x is almost homogeneous at the origin then x can be embedded in a euclidean space in an almost bi lipschitz way

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

the intrinsic hypoelliptic laplacian and its heat kernel on unimodular lie groups we present an invariant definition of the hypoelliptic laplacian on sub riemannian structures with constant growth vector using the popp s volume form introduced by montgomery this definition generalizes the one of the laplace beltrami operator in riemannian geometry in the case of left invariant problems on unimodular lie groups we prove that it coincides with the usual sum of squares we then extend a method first used by hulanicki on the heisenberg group to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular lie group of type i the main tool is the noncommutative fourier transform we then study some relevant cases su 2 so 3 sl 2 with the metrics inherited by the killing form and the group se 2 of rototranslations of the plane our study is motivated by some recent results about the cut and conjugate loci on these sub riemannian manifolds the perspective is to understand how singularities of the sub riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation

expanders and box spaces we consider box spaces of finitely generated residually finite groups g and try to distinguish them up to coarse equivalence we show that for n 2 the group sln z has a continuum of box spaces which are pairwise non coarsely equivalent expanders moreover varying the integer n 3 expanders given as box spaces of sln z are pairwise inequivalent similarly varying the prime p expanders given as box spaces of sl2 z p are pairwise inequivalent a strong form of non expansion for a box space is the existence of 0 1 such that the diameter of each component xn satisfies diam xn xn by bt15 the existence of such a box space implies that g virtually maps onto z we establish the converse for the lamplighter group z 2z z and for a semi direct product z 2 z such box spaces are explicitly constructed using specific congruence subgroups we finally introduce the full box space of g i e the coarse disjoint union of all finite quotients of g we prove that the full box space of a group mapping onto the free group f2 is not coarsely equivalent to the full box space of an s arithmetic group satisfying the congruence subgroup property

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

groups quasi isometric to complex hyperbolic space we show that any finitely generated group quasi isometric to complex hyperbolic space is a finite extension of a properly discontinuous cocompact subgroup of the isometry group

new properties of convex functions in the heisenberg group we prove some new properties of the weakly h convex functions recently introduced by danielli garofalo and nhieu as an interesting application of our results we prove a theorem of busemann feller alexandrov type in the heisenberg groups h n 1 2

gibbs phenomenon for fourier partial sums on zp abstract it is of interest to know whether the gibbs phenomenon occurs on a local field a p adic heaviside function on the group of p adic integers is defined as an analogy of real variable heaviside function and it is also shown that there exists the gibbs phenomenon with an undershoot of at least 1 p 1 as an approximate level as a consequence of the theorem we see that the fourier partial sum for a heaviside function does not converge at the point of discontinuity

weak fubini property and infinity harmonic functions in riemannian and sub riemannian manifolds we examine the relationship between infinity harmonic functions absolutely minimizing lipschitz extensions strong absolutely minimizing lipschitz extensions and absolutely gradient minimizing extensions in carnot carath eodory spaces using the weak fubini property we show that absolutely minimizing lipschitz extensions are infinity harmonic in any sub riemannian manifold

filling invariants at infinity for manifolds of nonpositive curvature in this paper we construct and study isoperimetric functions at infinity for hadamard manifolds these quasi isometry invariants give a measure of the spread of geodesics in such a manifold

large scale conformal rigidity in dimension three let x be a noncompact smooth manifold two complete riemannian metrics on x in a given conformal class can have very different asymptotic geometries for instance starting with the euclidean plane e2 with polar coordinates r multiplying the euclidean metric g0 dr2 r2d 2 by a function equal to 1 r2 outside a compact neighborhood of the origin one obtains a complete riemannian metric g which is quasi isometric to a half line the metrics g and g0 are in the same conformal class but they are not quasi isometric in fact they have different asymptotic dimensions one can ask whether something similar can happen to two finitely generated groups and 0 can they be coarsely quasi conformal in some sense and yet not quasi isometric to give a precise meaning to this question we will choose riemannian manifoldsx0 x that are geometric models for 0 and consider conformal mappings between x0 and x first let us recall some definitions two metric spaces x1 d1 and x2 d2 are quasi isometric if there are constants 1 and c 0 and a map f x1 x2 satisfying 1 d1 x x c d2 f x f x d1 x x c x x x1 y x2 x x1 d f x y c

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

orbit equivalence rigidity and bounded cohomology we establish new results and introduce new methods in the theory of measurable orbit equivalence using bounded cohomology of group representations our rigidity statements hold for a wide uncountable class of groups arising from negative curvature geometry amongst our applications are a measurable mostow type rigidity theorems for products of negatively curved groups b prime factorization results for measure equivalence c superrigidity for orbit equivalence d the first examples of continua of type ii1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic

sub riemannian geometry of the coefficients of univalent functions abstract we consider coefficient bodies m n for univalent functions based on the lowner kufarev parametric representation we get a partially integrable hamiltonian system in which the first integrals are kirillov s operators for a representation of the virasoro algebra then m n are defined as sub riemannian manifolds given a lie poisson bracket they form a grading of subspaces with the first subspace as a bracket generating distribution of complex dimension two with this sub riemannian structure we construct a new hamiltonian system to calculate regular geodesics which turn to be horizontal lagrangian formulation is also given in the particular case m 3

lipschitz continuity aleksandrov theorem and characterizations for h convex functions in the geometries of stratified groups we show that h convex functions locally bounded from above are locally lipschitz continuous and that the class of v convex functions exactly corresponds to the class of upper semicontinuous h convex functions as a consequence v convex functions are locally lipschitz continuous in every stratified group in the class of step 2 groups we characterize locally lipschitz h convex functions as measures whose distributional horizontal hessian is positive semidefinite in euclidean space the same results were obtained by dudley and reshetnyak we prove that a continuous h convex function is a e twice differentiable whenever its second order horizontal derivatives are radon measures

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

detection of edges in spectral data ii nonlinear enhancement we discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities where f x f x f x neq 0 our approach is based on two main aspects localization using appropriate concentration kernels and separation of scales by nonlinear enhancement to detect such edges one employs concentration kernels k epsilon cdot depending on the small scale epsilon it is shown that odd kernels properly scaled and admissible in the sense of having small w 1 infty moments of order cal o epsilon satisfy k epsilon f x f x cal o epsilon thus recovering both the location and amplitudes of all edges as an example we consider general concentration kernels of the form k sigma n t sum sigma k n sin kt to detect edges from the first 1 epsilon n spectral modes of piecewise smooth f s here we improve in generality and simplicity over our previous study in a gelb and e tadmor appl comput harmon anal 7 1999 pp 101 135 both periodic and nonperiodic spectral projections are considered we identify in particular a new family of exponential factors sigma exp cdot with superior localization properties the other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges where k epsilon f x sim f x neq 0 and the smooth regions where k epsilon f cal o epsilon sim 0 numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors

commensurability of groups quasi isometric to raags let g be a right angled artin group with defining graph gamma and let h be a finitely generated group quasi isometric to g we show if g satisfies that 1 its outer automorphism group is finite 2 gamma does not contain any induced 4 cycles 3 gamma is star rigid then h is commensurable to g we show condition 2 is sharp in the sense that if gamma contains an induced 4 cycle then there exists an h quasi isometric to g but not commensurable to g moreover one can drop condition 1 if h is a uniform lattice acting on the universal cover of the salvetti complex of g as a consequence we obtain a conjugation theorem for such uniform lattices the ingredients of the proof include a blow up building construction in huang and kleiner duke math j 167 3 537 602 2018 https doi org 10 1215 00127094 2017 0042 and a haglund wise style combination theorem for certain class of special cube complexes however in most of our cases relative hyperbolicity is absent so we need new ingredients for the combination theorem

hamiltonian b series and a lie algebra of non rooted trees the conditions for a b series to be a hamiltonian vector field imply that it may be given as a series indexed by free trees i e trees without root at present the pre lie structure of rooted trees plays an important role in the study of numerical methods for ordinary differential equations as does the associated lie bracket on rooted trees obtained by antisymmetrization we give a substitute for this lie bracket defined on free trees that reflects the lie bracket of hamiltonian b series and illustrate an application of this to the backward error analysis of symplectic numerical integrators

anti de sitter strictly ghc regular groups which are not lattices for d 4 5 6 7 8 we exhibit examples of mathrm ads d 1 strictly ghc regular groups which are not quasi isometric to the hyperbolic space mathbb h d nor to any symmetric space this provides a negative answer to question 5 2 in 9a12 and disproves conjecture 8 11 of barbot merigot bm12 we construct those examples using the tits representation of well chosen coxeter groups on the way we give an alternative proof of moussong s hyperbolicity criterion mou88 for coxeter groups built on danciger gueritaud kassel dgk17 and find examples of coxeter groups w such that the space of strictly ghc regular representations of w into mathrm po d 2 mathbb r up to conjugation is disconnected

extremal elements in coxeter groups and metric commensurators of kac moody groups we prove a characterization of irreducible non spherical and non affine coxeter groups motivated by applications to metric commensurators of kac moody groups and twinnings at finite distance

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

estimates on the markov convexity of carnot groups and quantitative nonembeddability abstract we show that every graded nilpotent lie group g of step r equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading this includes all carnot groups with carnot caratheodory metric is markov p convex for all p 2 r we also show that this is sharp whenever g is a carnot group with r 3 a free carnot group or a jet space group such groups are not markov p convex for any p 0 2 r this continues a line of research started by li who proved this sharp result when g is the heisenberg group as corollaries we obtain new estimates on the non bilipschitz embeddability of some finitely generated nilpotent groups into nilpotent lie groups of lower step sharp estimates of this type are known when the domain is the heisenberg group and the target is a uniformly convex banach space or l 1 but not when the target is a nonabelian nilpotent group

compression bounds for lipschitz maps from the heisenberg group to l 1 we prove a quantitative bi lipschitz non embedding theorem for the heisenberg group with its carnot caratheodory metric and apply it to give a lower bound on the integrality gap of the goemans linial semidefinite relaxation of the sparsest cut problem

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

canonical b series b series provide a powerful general tool to express numerical methods for differential equations many differential equations are of hamiltonian form and there has been much recent interest in constructing so called canonical or symplectic integrators for the hamiltonian case in this paper we provide a necessary and sufficient condition for a b series to correspond to a canonical method

symmetric patterns of geodesics and automorphisms of surface groups we prove a non equivariant version of mostow rigidity for symmetric patterns of geodesics in hyperbolic space this result allows for a classification of pseudo anosov surface group automorphisms based on the large scale geometric structure of their orbits

gromov s measure equivalence and rigidity of higher rank lattices in this paper the notion of measure equivalence me of countable groups is studied me was introduced by gromov as a measure theoretic analog of quasi isometries all lattices in the same locally compact group are measure equivalent this is one of the motivations for this notion the main result of this paper is me rigidity of higher rank lattices any countable group which is me to a lattice in a simple lie group g of higher rank is commensurable to a lattice in g

singular integrals on self similar sets and removability for lipschitz harmonic functions in heisenberg groups in this paper we study singular integrals on small that is measure zero and lower than full dimensional subsets of metric groups the main examples of the groups we have in mind are euclidean spaces and heisenberg groups in addition to obtaining results in a very general setting the purpose of this work is twofold we shall extend some results in euclidean spaces to more general kernels than previously considered and we shall obtain in heisenberg groups some applications to harmonic in the heisenberg sense functions of some results known earlier in euclidean spaces

quasi isometric classification of graph manifold groups we show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi isometric we also classify the quasi isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs a corollary is the quasi isometric classification of artin groups whose presentation graphs are trees in particular any two right angled artin groups whose presentation graphs are trees of diameter greater than 2 are quasi isometric further this quasi isometry class does not include any other right angled artin groups a finitely generated group can be considered geometrically when endowed with a word metric up to quasi isometric equivalence such metrics are unique in this article only finitely generated groups are considered given a collection of groups g gromov 11 proposed the fundamental questions of identifying which groups are quasi isometric to those in g rigidity and which groups in g are quasi isometric to each other classification in this article we focus on the classification question for graph manifold groups and right angled artin groups a compact 3 manifold m is called geometric if m m admits a geometric structure in the sense of thurston i e a complete locally homogeneous riemannian metric of finite volume thurston s geometrization conjecture see 29 17 21 23 22 provides that every irreducible 3 manifold of zero euler characteristic i e with boundary consisting only of tori and klein bottles admits a decomposition along tori and klein bottles into geometric pieces the minimal such decomposition being called the geometric decomposition there is a considerable literature on quasi isometric rigidity and classification of 3 manifold groups the rigidity results can be briefly summarized in the following form

a stability barrier for reconstructions from fourier samples we prove that any stable method for resolving the gibbs phenomenon that is recovering high order accuracy from the first m fourier coefficients of an analytic and nonperiodic function can converge at best root exponentially fast in m any method with faster convergence must also be unstable and in particular exponential convergence implies exponential ill conditioning this result is analogous to a recent theorem of platte trefethen and kuijlaars concerning recovery from pointwise function values on an equispaced m grid the main step in our proof is an estimate for the maximal behavior of a polynomial of degree n with bounded m term fourier series this estimate is related to a conjecture of hrycak and grochenig in the final part of the paper we discuss the implications of our main theorem to polynomial based interpolation and least squares approaches for overcoming the gibbs phenomenon

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

boolean percolation on doubling graphs we consider the discrete boolean model of percolation on weighted graphs satisfying a doubling metric condition we study sufficient conditions on the distribution of the radii of balls placed at the points of a bernoulli point process for the absence of percolation provided that the parameter of the underlying point process is small enough we exhibit three families of interesting graphs where the main result of this work holds finally we give sufficient conditions for ergodicity of the discrete boolean model of percolation

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

higher rank hyperbolicity the large scale geometry of hyperbolic metric spaces exhibits many distinctive features such as the stability of quasi geodesics the morse lemma the visibility property and the homeomorphism between visual boundaries induced by a quasi isometry we prove a number of closely analogous results for spaces of rank n ge 2 in an asymptotic sense under some weak assumptions reminiscent of nonpositive curvature for this purpose we replace quasi geodesic lines with quasi minimizing locally finite n cycles of r n volume growth prime examples include n cycles associated with n quasiflats solving an asymptotic plateau problem and producing unique tangent cones at infinity for such cycles we show in particular that every quasi isometry between two proper text cat 0 spaces of asymptotic rank n extends to a class of n 1 cycles in the tits boundaries

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

singular solutions homogeneous norms and quasiconformal mappings in carnot groups in any carnot nilpotent stratified lie group g of homogeneous dimension q green s function u for the q laplace equation exists and is unique we prove that there exists a constant gamma gamma g so that n e gamma u is a homogeneous norm in g then the extremal lengths of spherical ring domains measured with respect to n can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the carnot caratheodory metric applications include regularity properties of quasiconformal mappings and a geometric characterization of bi lipschitz mappings

sub riemannian calculus on hypersurfaces in carnot groups we develop a sub riemannian calculus for hypersurfaces in graded nilpotent lie groups we introduce an appropriate geometric framework such as horizontal levi civita connection second fundamental form and horizontal laplace beltrami operator we analyze the relevant minimal surfaces and prove some basic integration by parts formulas using the latter we establish general first and second variation formulas for the horizontal perimeter in the heisenberg group such formulas play a fundamental role in the sub riemannian bernstein problem

a characterization of cocompact hyperbolic and finite volume hyperbolic groups in dimension three we show that a cocompact hyperbolic group in dimension 3 is char acterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space we prove a similar theorem for finite volume hyperbolic groups in dimension 3 a group g is called cocompact hyperbolic of dimension n if and only if it can act isometrically properly discontinuously and cocompactly on hyperbolic space h proper discontinuity implies that the action has at most finite kernel our main result states that a group g is cocompact hyperbolic of dimension 3 if and only if it has a finite generating set c and associated cayley graph f r cr c such that the word metric on t is comparable in the large quasi isometric with the standard hyperbolic metric on h3 we have a similar result for finite volume hyperbolic groups in dimension 3 we believe that our results generalize to all dimensions at least in the cocompact case our result for groups of finite volume employs a different metric space instead of the cayley graph with word metric we use an augmented cayley graph associated with a group g and a finite family of euclidean subgroups of g we prove that the group acts isometrically and properly discontinuously on h3 with orbit space of finite volume if and only if the augmented cayley graph is quasi isometric with h3 precise definitions of all of these terms will appear in 2 the proof of the cocompact case appears in 3 the proof of the finite volume case appears in 4 our results are similar to a result of gromov gl but without the added assumption that the group be the fundamental group of a convex path space there are other results on the classification of groups up to quasi isometry in g2 in particular a group is quasi isometric to e if and only if there is a subgroup z of finite index it would be interesting to know which groups are quasi isometric to the 3 dimensional geometries h2 x e1 solv and psl 2 r gromov s work on groups with polynomial growth g3 bk implies that a group is quasi isometric to nil if and only if it has a subgroup of finite index which is the fundamental group of a compact nil 3 orbifold notice that all compact

quasi actions on trees i bounded valence given a bounded valence bushy tree t we prove that any cobounded quasi action of a group g on t is quasiconjugate to an action of g on another bounded valence bushy tree t this theorem has many applications quasi isometric rigidity for fundamental groups of finite bushy graphs of coarse pd n groups for each fixed n a generalization to actions on cantor sets of sullivan s theorem about uniformly quasiconformal actions on the 2 sphere and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice finally we give the first examples of two finitely generated groups which are quasi isometric and yet which cannot act on the same proper geodesic metric space properly discontinuously and cocompactly by isometries

lipschitz algebras and derivations ii exterior differentiation abstract basic aspects of differential geometry can be extended to various non classical settings lipschitz manifolds rectifiable sets sub riemannian manifolds banach manifolds wiener space etc although the constructions differ in each of these cases one can define a module of measurable 1 forms and a first order exterior derivative we give a general construction which applies to any metric space equipped with a finite measure and produces the desired result in all of the above cases it agrees with cheeger s construction when the latter is defined it also applies to an important class of dirichlet spaces where however the known first order differential calculus in general differs from ours although the two are related

on inverse methods for the resolution of the gibbs phenomenon when fourier expansions or more generally spectral methods are used for the representation of nonsmooth functions then one has to face the so called gibbs phenomenon considerable progresses have been made these last years to overcome the gibbs phenomenon using direct or inverse approaches both in the discrete or continuous framework a discrete inverse method for the global or local reconstruction of a non smooth function starting from its oscillating trigonometric polynomial interpolant is introduced and both its capabilities and limits are emphasized

groups quasi isometric to right angled artin groups we characterize groups quasi isometric to a right angled artin group raag g with finite outer automorphism group in particular all such groups admit a geometric action on a cat 0 cube complex that has an equivariant fibering over the davis building of g this characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi isometric to certain raags

coarse differentiation and quantitative nonembeddability for carnot groups abstract we give lower bound estimates for the macroscopic scale of coarse differentiability of lipschitz maps from a carnot group with the carnot caratheodory metric g d c c to a few different classes of metric spaces using this result we derive lower bound estimates for quantitative nonembeddability of lipschitz embeddings of g into a metric space x d x if x is either an alexandrov space with nonpositive or nonnegative curvature a superreflexive banach space or another carnot group that does not admit a bilipschitz homomorphic embedding of g for the same targets we can further give lower bound estimates for the bilipschitz distortion of every embedding f b n x where b n is the ball of radius n of a finitely generated nonabelian torsion free nilpotent group g we also prove an analogue of bourgain s discretization theorem for carnot groups and show that carnot groups have nontrivial markov convexity these give the first examples of metric spaces that have nontrivial markov convexity but cannot bilipschitzly embed into banach spaces of nontrivial markov convexity

group splittings and asymptotic topology it is a consequence of the theorem of stallings on groups with many ends that splittings over finite groups are preserved by quasi isometries in this paper we use asymptotic topology to show that group splittings are preserved by quasi isometries in many cases roughly speaking we show that splittings are preserved under quasi isometries when the vertex groups are fundamental groups of aspherical manifolds or more generally coarse pd n groups and the edge groups are smaller than the vertex groups

a notable family of entire intrinsic minimal graphs in the heisenberg group which are not perimeter minimizing one of the main objectives of this paper is to unravel a new interesting phenomenon of the sub riemannian bernstein problem with respect to its euclidean ancestor with the purpose of also indicating a possible line of attack toward its solution we show that the global intrinsic graphs 1 2 are unstable critical points of the horizontal perimeter as a consequence of this fact the study of the stability acquires a central position in the problem itself

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

quasisymmetric maps on the boundary of a negatively curved solvable lie group we describe all the self quasisymmetric maps on the ideal boundary of a particular negatively curved solvable lie group as applications we derive some rigidity properties for quasiisometries of the solvable lie group

recognizing constant curvature discrete groups in dimension 3 we characterize those discrete groups g which can act properly discontinuously isometrically and cocompactly on hyperbolic 3 space ihi3 in terms of the combinatorics of the action of g on its space at infinity the major ingredients in the proof are the properties of groups that are negatively curved in the large that is gromov hyperbolic the combinatorial riemann mapping theorem and the sullivan tukia theorem on groups which act uniformly quasiconformally on the 2 sphere

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

hardy littlewood inequality for quasiregular maps on carnot groups abstract we generalize a hardy littlewood inequality for l p norms of conjugate harmonic functions to horizontal components of quasiregular mappings on carnot groups

a variant of gromov s problem on holder equivalence of carnot groups abstract it is unknown if there exists a locally holder homeomorphism f r 3 h 1 for any 1 2 2 3 although the identity map r 3 h 1 is locally 1 2 holder more generally gromov asked given k and a carnot group g for which does there exist a locally holder homeomorphism f r k g here we equip a carnot group g with the carnot caratheodory metric in 2014 balogh hajlasz and wildrick considered a variant of this problem these authors proved that if k n there does not exist an injective 1 2 holder mapping f r k h n that is also locally lipschitz as a mapping into r 2 n 1 for their proof they use the fact that h n is purely k unrectifiable for k n in this paper we will extend their result from the heisenberg group to model filiform groups and carnot groups of step at most three we will now require that the carnot group is purely k unrectifiable the main key to our proof will be showing that 1 2 holder maps f r k g that are locally lipschitz into euclidean space are weakly contact proving weak contactness in these two settings requires understanding the relationship between the algebraic and metric structures of the carnot group we will use coordinates of the first and second kind for carnot groups

sub riemannian calculus and monotonicity of the perimeter for graphical strips we consider the class of minimal surfaces given by the graphical strips mathcal s in the heisenberg group mathbb h 1 and we prove that for points p along the center of mathbb h 1 the quantity frac sigma h mathcal s cap b p r r q 1 is monotone increasing here q is the homogeneous dimension of mathbb h 1 we also prove that these minimal surfaces have maximum volume growth at infinity

symplectic methods based on decompositions we consider methods that integrate systems of differential equations dy dt f y by taking advantage of a decomposition of the right hand side f sum f nu we derive a general necessary and sufficient condition for those methods to be symplectic for hamiltonian problems special attention is given to the case of additive runge kutta methods

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws