a proof that all seifert 3 manifold groups and all virtual surface groups are conjugacy separable abstract we prove that the fundamental group of any seifert 3 manifold is conjugacy separable that is conjugates may be distinguished in finite quotients or equivalently conjugacy classes are closed in the pro finite topology

algebraic convergence of schottky groups a discrete faithful representation of the free group on g generators f g into isom h 3 is said to be a schottky group if h 3 d is homeomorphic to a handlebody h g where d is the domain of discontinuity for s action on the sphere at infinity for h 3 schottky space s g the space of all schottky groups is parameterized by the quotient of the teichmuller space f s g of the closed surface of genus g by mod 0 h g where mod 0 h g is the group of isotopy classes of homeomorphisms of s g which extend to homeomorphisms of h g which are homotopic to the identity masur exhibited a domain o h g of discontinuity for mod 0 h g s action on pl s g the space of projective measured laminations on s g so b h g o h g mod 0 h g may be appended to s g as a boundary

finite index subgroups of mapping class groups let g 3 and n 0 and let mg n be the mapping class group of a surface of genus g with n boundary components we prove that mg n contains a unique subgroup of index 2g 1 2g 1 up to conjugation a unique subgroup of index 2g 1 2g 1 up to conjugation and the other proper subgroups ofmg n are of index greater than 2g 1 2g 1 in particular the minimum index for a proper subgroup of mg n is 2g 1 2g 1 ams subject classification primary 57m99 secondary 20g40 20e28 0 introduction and statement of results the interaction between mapping class groups and finite groups has long been a topic of interest the famous hurwitz bound of 1893 showed that the mapping class group of a closed riemann surface of genus g has an upper bound of 84 g 1 for the order of its finite subgroups and kerckhoff showed that the order of finite cyclic subgroups is bounded above by 4g 2 19 20 the subject of finite index subgroups of mapping class groups was brought into focus by grossman s discovery that the mapping class group mg n 0 homeo g n of an oriented surface g n of genus g and n boundary components is residually finite and thus well endowed with subgroups of finite index 15 homeo g n denotes the space of those homeomorphisms of g n that preserve the orientation and are the identity on the boundary this prompts the dual question what is the minimum index mi mg n of a proper subgroup of finite index in mg n results to date have suggested that like the maximum finite order question the minimum index question should have an answer that is linear in g the best previously published bound is mi mg n 4g 4 for g 3 see 26 this inequality is used by aramayona and souto to prove that if g 6 and g 2g 1 then any nontrivial homomorphism mg n mg n is induced by an embedding 1 it is also an important ingredient in the proof of zimmermann 33 that for g 3 and 4 the minimal nontrivial quotient of mg 0 is sp2g f2 the headline result of this paper is the following exact exponential bound theorem 0 1 for g 3 and n 0 mi mg n mi sp2g z mi sp2g f2 2g 1 2g 1 this exponential bound is all the more surprising since in similar questions we get linear expected bounds for instance bridson 5 6 has proved that a mapping class group of a surface

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

doubly nonlinear thin film equations in one space dimension we consider a free boundary problem for a class of fourth order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative the problem is relevant in the description of the surface tension driven spreading of a non newtonian liquid over a solid surface in the complete wetting regime relying solely on global and local energy estimates and on bernis inequalities we prove existence of solutions to this problem and obtain sharp upper bounds for the propagation of their support a necessary condition for the occurrence of waiting time phenomena is also derived

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

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

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

subvarieties of moduli space determined by finite groups acting on surfaces suppose the finite group g acts as orientation preserving homeomorphisms of the oriented surface s of genus g this determines an irreducible subvariety m g g of the moduli space m g of riemann surfaces of genus g consisting of all surfaces with a group g 1 of holomorphic homeomorphisms of the same topological type as g this family is determined by an equivalence class of epimorphisms from a fuchsian group to g whose kernel is isomorphic to the fundamental group of s to examine the singularity of m g along this family one needs to know the full automorphism group of a generic element of m g g

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

braid groups and the co hopfian property abstract let b n be the braid group on n 4 strands we prove that b n modulo its center is co hopfian we then show that any injective endomorphism of b n is geometric in the sense that it is induced by a homeomorphism of a punctured disk we further prove that any injection from b n to b n 1 is geometric for n 7 additionally we obtain analogous results for mapping class groups of punctured spheres the methods use thurston s theory of surface homeomorphisms and build upon work of ivanov mccarthy

droplet spreading intermediate scaling law by pde methods we consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity the standard lubrication approximation leads to an evolution equation for the film height h that is ill posed when the spreading is limited by the no slip boundary condition at the liquid solid interface due to a singularity at the moving contact line the most common relaxation of the no slip boundary condition removes this singularity but introduces a new physical length scale the slippage length b it is believed that this microscopic length scale only enters logarithmically in the effective that is macroscopic spreading behavior in this paper we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b more precisely we prove a scaling law for the diameter of the apparent that is macroscopic support of the droplet in time this is an intermediate scaling law it takes an initial layer to forget the initial droplet shape whereas after a long time the droplet is so thin that its spreading is governed by the physics on the scale b our proof works by deriving suitable estimates for physically relevant integral quantities the free energy the length of the apparent support and their respective rates of change as opposed to matched asymptotic methods this pde approach closely mimics a simple heuristic argument based on the gradient flow structure 2002 john wiley sons inc

low dimensional lattice groups for the continuum mechanics of phase transitions in crystals first i review and adapt some classical and recent formulations of problems relating to the classification of two and three dimensional lattices then i use these methods to calculate lattice groups that are of interest in the continuum mechanics of phase transitions in crystals and to construct invariant neighbourhoods that are useful in the same context the paper is designed to provide a perspective of similar calculations which employ methods of traditional crystallography

the deformation spaces of convex rp structures on 2 orbifolds we determine that the deformation space of convex real projective structures that is projectively flat torsion free connections with the geodesic convexity property on a compact 2 orbifold of negative euler characteristic is homeomorphic to a cell of certain dimension the basic techniques are from thurston s lecture notes on hyperbolic 2 orbifolds the previous work of goldman on convex real projective structures on surfaces and some classical geometry

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

optimal lower bounds on asymptotic support propagation rates for the thin film equation abstract we derive lower bounds on asymptotic support propagation rates for strong solutions of the cauchy problem for the thin film equation the bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp our results hold in case of at most three spatial dimensions and n 1 2 92 the result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality combined with some entropy production estimates the optimal rate of propagation is obtained to the best of our knowledge these are the first lower bounds on asymptotic support propagation rates for higher order nonnegativity preserving parabolic equations

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

material symmetry and singularities in solids we examine the relations between the symmetry group of a crystalline or solid body and the line singularities in the deformation or microstructural field which the material can support a method is developed which in some relevant cases allows us to describe a singularity by means of a suitable derham current supported on the defect submanifold this current measures the obstruction to the extension of the local deformation or microstructure to a globally defined field the procedure described here yields the basis of a formulation of weak field equations for a defective material and clarifies the structure of the compatibility relations for virtually any class of material symmetry

an upper bound for the waiting time for nonlinear degenerate parabolic equations an upper bound is obtained for the time when the support of the solution of some nonlinear degenerate parabolic equations begins to spread solutions to many nonlinear degenerate diffusion equations which are compactly supported initially remain so at future times see 4 7 8 we obtain here an upper bound for the time when the support begins to spread the so called waiting time in one space dimension such an upper bound was found by b f knerr 7 who derived a weak ordinary differential equation along the interface the boundary of the support and used it to show that the waiting time depends on the size of the initial data near the interface see also 2 4 11 12 we replace the equation along the interface with a first order differential in equality for a local average of the solution near a point on the interface this allows estimation of the waiting time in higher dimensions while in one dimension it reproduces previous results 4 7 in the particular case of the porous medium equation similar results can also be obtained from a new and sharp harnack inequality due to d g aronson and l a caffarelli 3 n d alikakos 1 has recently supplied a necessary and sufficient condition for zero waiting time in higher dimensions which relies in part upon this inequality nevertheless even for the case of the porous medium equation our results are elementary and self contained finally we would like to thank j l vasquez for an informative conversation with the second author and for suggesting that we seek a local version of our preliminary results 1 preliminaries let uq and g be nonnegative functions a function u x t is said to be a weak solution of the nonlinear degenerate parabolic equation ut ag u u x 0 uo x if for some t 0 u satisfies

braid groups and discrete diffeomorphisms of the punctured disk we show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand as an application of this method we also prove that there is no cohomological obstruction to lifting the standard embedding mathrm br 2g 2 hookrightarrow mathrm mod g 2 to a group homomorphism between diffeomorphism groups

conjugacy in garside groups iii periodic braids abstract an element in artin s braid group b n is said to be periodic if some power of it lies in the center of b n in this paper we prove that all previously known algorithms for solving the conjugacy search problem in b n are exponential in the braid index n for the special case of periodic braids we overcome this difficulty by putting to work several known isomorphisms between garside structures in the braid group b n and other garside groups this allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms this paper is the third in a series of papers by the same authors about the conjugacy problem in garside groups they have a unified goal the development of a polynomial algorithm for the conjugacy decision and search problems in b n which generalizes to other garside groups whenever possible it is our hope that the methods introduced here will allow the generalization of the results in this paper to all artin tits groups of spherical type

low dimensional lattices vi voronoi reduction of three dimensional lattices the aim of this paper is to describe how the voronoi cell of a lattice changes as that lattice is continuously varied the usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice the present paper deals with dimensions n leq 3 a sequel will treat four dimensional lattices an elegant algorithm is given for the voronoi reduction of a three dimensional lattice leading to a new proof of voronoi s theorem that every lattice of dimension n leq 3 is of the first kind and of fedorov s classification of the three dimensional lattices into five types there is a very simple formula for the determinant of a three dimensional lattice in terms of its conorms

thin film equations with partial wetting energy existence of weak solutions abstract the capillarity driven evolution with slip of a thin liquid film over a dry surface is considered in the regime of partial wetting the focus is on the simplest model case of a constant non zero dynamic contact angle in the lubrication approximation for the analytical treatment of the corresponding free boundary problem a new strategy is proposed based on the introduction of an ad hoc class of disjoining pressures which tend to concentrate at triple junctions a first investigation of this approach yields the existence of weak solutions which satisfy the dissipation relation as an inequality and which are different from those with zero contact angle a heuristic argument is also presented in order to clarify the connection between contact angle and dissipation relation it shows that moving droplets which satisfy the dissipation relation as an equality are forced to have the prescribed contact angle

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

small and waiting time behavior of the thin film equation we consider the small time behavior of interfaces of zero contact angle solutions to the thin film equation for a certain class of initial data through asymptotic analyses we deduce a wide variety of behavior for the free boundary point these are supported by extensive numerical simulations

nielsen realization for 3 manifolds containing two sided projective planes for compact irreducible sufficiently large 3 manifolds containing 2 sided projective planes we consider the following realization problem given a finite subgroup of the outer automorphism group of the fundamental group is there a finite group of homeomorphisms which induces this subgroup

blow up and extinction for a thin film equation with initial boundary value conditions abstract in this paper we consider an initial boundary value problem for a thin film equation with the nonlocal source u q u q d x in a bounded domain due to the comparison principle being invalid we use the potential well method to give a threshold result of global existence and non existence for the sign changing weak solutions further we obtain the conditions under which the global solutions extinct in finite time and study the asymptotic behavior of the non extinct solutions

a note on the model of crystalline defects in ericksen pitteri neighborhoods abstract we apply here the topological theory of defects to a simple model of crystalline microstructure appropriate to the study of elastic crystals the model is obtained by restricting the range of the map describing the local lattice to the neighborhood of a given reference bravais lattice this set may be described as the quotient of a neighborhood of the identity in gl r3 the ericksen pitteri neighborhood by a finite subgroup of the symmetry group of the reference lattice containing the point group we show that the topological structure of this set is sufficiently rich to describe disclinations related to point group symmetries and twins related to non rotational symmetries but excludes defects corresponding to simple shear symmetries intuitively related to plastic deformations we also discuss the dependence of the description from the choice of the reference lattice

upper bounds on waiting times for the thin film equation the case of weak slippage we derive upper bounds on the waiting time of solutions to the thin film equation in the regime of weak slippage n in 2 frac 32 11 in particular we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary for n in 2 frac 32 11 our estimates are sharp for n 2 they are sharp up to a logarithmic correction term note that the case n 2 corresponds with a grain of salt to the assumption of the navier slip condition at the fluid solid interface we also obtain results in the regime of strong slippage n in 1 2 however in this regime we expect them not to be optimal our method is based on weighted backward entropy estimates hardy s inequality and singular weight functions we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long

the modulus of continuity for γ0 m semi classical limits we study the behavior of a large eigenvalue limit of eigenfunctions for the hyperbolic laplacian for the modular quotient sl 2 fejer summation and results of s zelditch are used to show that the microlocal lifts of eigenfunctions have large eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle the limit is studied for hecke maass forms joint eigenfunctions of the hecke operators and the hyperbolic laplacian the first modulus of continuity result is presented for the limit the singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

euler characteristics and gysin sequences for group actions on boundaries let g be a locally compact group let x be a universal proper g space and let open image in new window be a g equivariant compactification of x that is h equivariantly contractible for each compact subgroup open image in new window let open image in new window assuming the baum connes conjecture for g with coefficients open image in new window and c x we construct an exact sequence that computes the map on k theory induced by the embedding open image in new window this exact sequence involves the equivariant euler characteristic of x which we study using an abstract notion of poincare duality in bivariant k theory as a consequence if g is torsion free and the euler characteristic open image in new window is non zero then the unit element of open image in new window is a torsion element of order open image in new window furthermore we get a new proof of a theorem of luck and rosenberg concerning the class of the de rham operator in equivariant k homology

a variational model for reconstructive phase transformations in crystals and their relation to dislocations and plasticity 1we study the reconstructive martensitic transformations in crystalline solids i e martensitic transformations in which the parent and product lattices have arithmetic symmetry groups admitting no finite supergroup the best known example of which is the bcc fcc transformation in iron we first describe the maximal ericksen pitteri neighborhoods in the space of lattice metrics thereby obtaining a quantitative characterization of the weak transformations which occur within these domains then focusing for simplicity on a two dimensional setting we construct a class of strain energy functions admitting large strains in their domain and which are invariant under the full symmetry group of the lattice in particular we exhibit an explicit energy suitable for the square to hexagonal reconstructive transformation in planar lattices we present a numerical scheme based on atomic scale finite elements and by means of our constitutive function we use it to analyze the effects of transformation cycling on a planar crystal this example illustrates the main phenomena related to the reconstructive martensitic phase changes in crystals in particular the formation of dislocations vacancies and interstitials in the lattice

characteristic classes of surface bundles let m be a c manifold if one wants to classify all differentiable fibre bundles over a given manifold which have m as fibres then one must study the topology of the corresponding structure group diffm the group of all diffeomorphisms of m equipped with the c topology it is easy to see that the connected component diffo m of the identity of diff m is an open subgroup therefore the natural topology on the quotient group m diff m diffo m the diffeotopy group of m is the discrete topology hence we have a fibration

arithmetic rigidity and units in group rings for any finite group g the group u z g of units in the integral group ring z g is an arithmetic group in a reductive algebraic group namely the zariski closure of sl1 q g in particular the isomorphism type of the q algebra q g determines the commensurability class of u z g we show that to a large extent the converse is true in fact subject to a certain restriction on the q representations of g the converse is exactly true

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation