thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

on the exponent of a finite group with an automorphism group of order twelve abstract suppose that a frobenius group fh with kernel f of order 4 and complement h of order 3 acts coprimely on a finite group g in such a manner that c g f 1 and c g h has exponent e we prove that the exponent of g is e bounded earlier results of this nature were known only for metacyclic frobenius groups

self bumping of deformation spaces of hyperbolic 3 manifolds let n be a hyperbolic 3 manifold and b a component of the interior of ah 1 n the space of marked hyperbolic 3 manifolds homotopy equivalent to n we will give topological conditions on n sufficient to give b such that for every sufficiently small neighbourhood v of v b is disconnected this implies that b is not a manifold with boundary

on centralizers in locally finite groups abstract the main result of the paper is the following theorem let g be a locally finite group containing a finite p subgroup a such that cg a is finite and a non cyclic subgroup b of order p2 such that cg b has finite exponent for all b b then g is almost locally solvable and has finite exponent

the large time error estimation of chebyshev pseudospectral approximation for a class of semilinear parabolic equations in this paper we use chebyshev pseudospectral method to analyze a class of semilinear parabolic equations we give out the error estimation between spectral approximate solution and exact solution on large time using local continuation method

the dihedral group as a group of automorphisms abstract suppose that d is a dihedral group generated by two involutions and let d act on a finite group g in such a manner that c g 1 we show that if c g and c g are both nilpotent of class c then g is nilpotent and the class of g is bounded solely in terms of c if both c g and c g are of exponent dividing e then the exponent of g is bounded solely in terms of e and d previously results of this kind were known only for groups acted on by frobenius groups of automorphisms

random walks kleinian groups and bifurcation currents let be a holomorphic family of representations of a finitely generated group g into psl 2 parameterized by a complex manifold we define a notion of bifurcation current in this context that is a positive closed current on describing the bifurcations of this family of representations in a quantitative sense it is the analogue of the bifurcation current introduced by demarco for holomorphic families of rational mappings on 1 our definition relies on the theory of random products of matrices so it depends on the choice of a probability measure on g

the long time behavior of finite element approximations of solutions of semilinear parabolic problems error estimates for finite element approximations of the solutions to semilinear parabolic problems are proved under the hypothesis that the exact solution is asymptotically stable as t to infty error estimates of optimal order that hold uniformly on the unbounded time interval 0 leqq t infty are obtained both semidiscrete and completely discrete approximations are considered

profinite groups and the fixed points of coprime automorphisms abstract the main result of the paper is the following theorem let q be a prime and a an elementary abelian group of order q 3 suppose that a acts coprimely on a profinite group g and assume that c g a is locally nilpotent for each a a then the group g is locally nilpotent

multiple bumping of components of deformation spaces of hyperbolic 3 manifolds let m be a hyperbolizable 3 manifold with nonempty incompressible boundary of negative euler characteristic suppose that b 1 b k is a collection of components of the interior of the space of complete marked hyperbolic 3 manifolds homotopy equivalent to m such that for any i j inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we prove that there is a geometrically finite hyperbolic structure on int m which is in the closure of each b i moreover we show that this structure can be constructed so as to admit quasiconformal deformations which also lie in the closure of every b i

slicing skinning and grafting we prove that a bers slice is never algebraic meaning that its zariski closure in the character variety has strictly larger dimension a corollary is that skinning maps are never constant the proof uses grafting and the theory of complex projective structures

bounding the exponent of a finite group with automorphisms a 4 where a s a 1 this follows immediately from the fact that for every prime p dividing g there is an a invariant sylow p subgroup p of g for a which even a stronger decomposition holds p s c a see for ag a p w x example 6 theorems 6 2 2 and 6 2 4 suppose that in this situation n is the least common multiple of the exponents of the centralizers c a g a g a we show that then the exponent of g is bounded by a number that depends only on n and on the group a

quasiconformal homeomorphisms and dynamics ii structural stability implies hyperbolicity for kleinian groups sous groupe de psl 2 c familles holomorphes d homeomorphismes de c lemme champs de ligne invariants familles holomorphes de groupes abstraitement isomorphes car finiment genere la stabilite structurelle implique geometriquement fini ou rigide theoreme a avec paraboles stabilite structurelle au sens de la dynamique differentiable

fixed points of coprime operator groups abstract let m be a positive integer and a an elementary abelian group of order q r with r 2 acting on a finite q group g we show that if for some integer d such that 2 d r 1 the dth derived group of c g a has exponent dividing m for any a a then g d has m q r bounded exponent and if r 1 c g a has exponent dividing m for any a a then r 1 g has m q r bounded exponent

long time behavior of the difference solutions to the 2d gks equation we study the long time behavior of the finite difference solution to the generalized kuramoto sivashinsky equation in two space dimensions with periodic boundary conditions the unique solvability of numerical solution is shown it is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d a h t a 0 finally we obtain the long time stability and convergence of the difference scheme our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems

prym varieties and teichmuller curves this paper gives a uniform construction of infinitely many primitive teichmuller curves v mg for g 2 3 and 4

on p groups of maximal class as groups of automorphisms abstract let p be a p group of maximal class and m be a maximal subgroup of p let be an element in p m such that c p p 2 and assume that p suppose that p acts on a finite group g in such a manner that c g m 1 we show that if c g is nilpotent then the fitting height of g is at most two and c g is contained in the fitting subgroup of g for p 2 without assuming that c g is nilpotent we prove that the fitting height h g of g is at most h c g 1 and the fitting series of c g coincides with the intersection of c g with the fitting series of g it is also proved that if c g x is of exponent dividing e for all elements x p m then the exponent of g is bounded solely in terms of e and p these results are in parallel with known results on action of frobenius and dihedral groups

on the discreteness criterion in sl 2 c in tukia and wang math scand 91 214 220 2002 the authors conjectured that a non elementary subgroup in sl 2 c containing parabolic and elliptic elements is discrete if and only if each two generator subgroup generated by a parabolic and an elliptic element is discrete the purpose of this paper is to give an affirmative answer to this question

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

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

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

on bounded conciseness of engel like words in residually finite groups abstract given a group word w and a group g the verbal subgroup w g is the one generated by all w values in g the word w is said to be boundedly concise if for each positive integer m there exists a number depending only on m and w bounding the order of w g whenever the set of w values in a group g has size at most m in the present article we show that various generalizations of the engel word are boundedly concise in residually finite groups

on pro p groups admitting a fixed point free automorphism an automorphism of a group g is called fixed point free fpf for short if the subgroup of fixed points cg x gyx x is trivial it is well known that existence of an fpf automorphism of a finite group g has a strong impact on the structure of g for example if is of prime order q then g is nilpotent 15 and the nilpotency class of g is bounded by some function of q 6 the minimal function h q with the above property is called the higman function also it follows from the classification of finite simple groups that if a finite group g admits an fpf automorphism such that g 1 then g is soluble 3 1 48 moreover by a result of th berger 1 the fitting height of g is bounded by the number of prime divisors of counting multiplicities however the structure of finite p groups admitting an fpf automorphism of arbitrary given order is still a mystery an unsolved problem is whether the derived length of such groups is bounded by some function of although there is varied evidence suggesting that this is so for instance by a theorem of v a kreknin 10 the derived length of any lie algebra admitting an fpf automorphism of order n is at most 2n 1 1 the problem remains unapproachable

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

positive laws in fixed points let a be an elementary abelian group of order at least q 3 acting on a finite q group g in such a manner that c g a satisfies a positive law of degree n for any a e a it is proved that the entire group g satisfies a positive lacy of degree bounded by a function of q and n only

on groups satisfying a positive law in fixed points abstract let a be an elementary abelian q group of order q 2 acting on a finite q group g in such a manner that the subgroup c g a c g b satisfies a positive law of degree n for any a b a it is proved that the entire group g satisfies a positive law of degree bounded by a function of q and n only