sufficient conditions for the filtration of a stationary processes to be standard let x be a stationary process with values in some sigma finite measured state space e mathcal e pi indexed by mathbb z call mathcal f x its natural filtration in 3 sufficient conditions were given for mathcal f x to be standard when e is finite and the proof used a coupling of all probabilities on the finite set e in this paper we construct a coupling of all laws having a density with regard to pi which is much more involved then we provide sufficient conditions for mathcal f x to be standard generalizing those in 3

on the bounded cohomology of semi simple groups s arithmetic groups and products we prove vanishing results for lie groups and algebraic groups over any local field in bounded cohomology the main result is a vanishing below twice the rank for semi simple groups related rigidity results are established for s arithmetic groups and groups over global fields we also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices

c c simplicity and the amenable radical a countable group is c simple if its reduced c algebra is simple it is well known that c simplicity implies that the amenable radical of the group must be trivial we show that the converse does not hold by constructing explicit counter examples we additionally prove that every countable group embeds into a countable group with trivial amenable radical and that is not c simple

crossed products extensions of lp bounds for amenable actions abstract we will extend earlier transference results due to neuwirth and ricard from the context of noncommutative l p spaces associated with amenable groups to that of noncommutative l p spaces associated with crossed products of amenable actions namely if m g c is a completely bounded fourier multiplier on l p then it extends to the crossed product with similar bounds provided that the action is amenable and trace preserving furthermore our construction also allows to extend g equivariant completely bounded operators acting on the space part to the crossed product provided that the generalized folner sets of the action satisfy certain accretivity property as a corollary we obtain stability results for maximal l p bounds over crossed products we derive using that stability results an application to the boundedness of smooth multipliers in the l p spaces of group algebras

multiple sign changing solutions to semilinear elliptic resonant problems abstract in this paper we use the topological degree theory and the critical groups to investigate the elliptic equation u f x u in and subject to u 0 on and establish a multiple solutions theorem which guarantees that this problem has at least six nontrivial solutions under some resonant conditions if this problem has only finitely many solutions then of these solutions there are two positive solutions two negative solutions and two sign changing solutions

regenerative representation for one dimensional gibbs states on montre que les etats de gibbs unidimensionnels peuvent etre representes comme des concatenations de listes infinies de mots i i d

operad groups and their finiteness properties abstract we propose a new unifying framework for thompson like groups using a well known device called operads and category theory as language we discuss examples of operad groups which have appeared in the literature before as a first application we prove a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type f this unifies and extends existing proofs that certain thompson like groups are of type f

nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups in this paper we consider a class of resonant cooperative elliptic systems based on some new results concerning the computations of the critical groups and the morse theory we establish some new results about the existence and multiplicity of solutions under new classes of conditions it turns out that our main results sharply improve some known results in the literature

homology of saddle point reduction and applications to resonant elliptic systems abstract in the setting of saddle point reduction we prove that the critical groups of the original functional and the reduced functional are isomorphic as application we obtain two nontrivial solutions for elliptic gradient systems which may be resonant both at the origin and at infinity the difficulty that the variational functional does not satisfy the palais smale condition is overcame by taking advantage of saddle point reduction our abstract results on critical groups are crucial

computations of critical groups and periodic solutions for asymptotically linear hamiltonian systems the purpose of this paper is two fold firstly we will give some parabolic like conditions which improve the well known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity as an application we then consider the second order hamiltonian systems u t h t u t 0 t r where h r rn r is t periodic in its first variable and h is asymptotically linear both at origin and at infinity based on the computations of the critical groups and the morse theory we obtain the existence and multiplicity results for periodic solutions under new classes of conditions it turns out that our main results improve sharply some known results in the literature

golod shafarevich groups with property t and kac moody groups we construct golod shafarevich groups with property t and thus provide counterexamples to a conjecture stated in a recent paper of zelmanov ze2 explicit examples of such groups are given by lattices in certain topological kac moody groups over finite fields we provide several applications of this result including examples of residually finite torsion non amenable groups

decreasing sequences of sigma fields and a measure change for brownian motion ii let f t t 0 be the filtration of a brownian motion b t t 0 on f p an example is given of a measure q p in the sense of absolute continuity for which f t t 0 is not the filtration of any brownian motion on f q this settles a 15 year old question

nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems abstract in this paper we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction q i t i 1 t q i 1 t q i t i t q i t q i 1 t i z where i t x i 2 x 2 v i t x is t periodic in t for t 0 and satisfies i n i for some n n q i t is the state of the i th particle assume that i 0 for some i z and v i t x denoting the derivative of v i respect to x is asymptotically linear with x both at origin and at infinity we would like to point out that this system is resonant both at origin and at infinity and not studied up to now based on some new results concerning the precise computations of the critical groups for a given m z we obtain the existence of nontrivial periodic solutions satisfying q i m n t t q i t for all t r and i z under some additional conditions

groups of piecewise projective homeomorphisms the group of piecewise projective homeomorphisms of the line provides straightforward torsion free counterexamples to the so called von neumann conjecture the examples are so simple that many additional properties can be established

integrable measure equivalence and rigidity of hyperbolic lattices we study rigidity properties of lattices in operatorname isom mathbf h n simeq mathrm so n 1 mathbb r n 3 and of surface groups in operatorname isom mathbf h 2 simeq mathrm sl 2 mathbb r in the context of integrable measure equivalence the results for lattices in operatorname isom mathbf h n n 3 are generalizations of mostow rigidity they include a cocycle version of strong rigidity and an integrable measure equivalence classification despite the lack of mostow rigidity for n 2 we show that cocompact lattices in operatorname isom mathbf h 2 allow a similar integrable measure equivalence classification

the dixmier problem lamplighters and burnside groups j dixmier asked in 1950 whether every non amenable group admits uniformly bounded representations that cannot be unitarised we provide such representations upon passing to extensions by abelian groups this gives a new characterisation of amenability furthermore we deduce that certain burnside groups are non unitarisable answering a question raised by g pisier

stationary processes whose filtrations are standard we study the standard property of the natural filtration associated to a 0 1 valued stationary process in our main result we show that if the process has summable memory decay then the associated filtration is standard we prove it by coupling techniques for a process whose associated filtration is standard we construct a product type filtration extending it based upon the usual couplings and the vershik s criterion for standardness

multiple sign changing solutions to the sturm liouville boundary value problem with resonance abstract in this paper we study the sturm liouville boundary value problem p x u q x u f x u x 0 1 subject to u 0 u 0 u 1 u 1 0 by constructing a new sobolev space h 1 0 1 we discuss the existence of multiple solutions especially the existence of multiple sign changing solutions to this problem when the nonlinear f is resonant both at 0 and by combining the methods of the morse theory the topological degree and the fixed point index we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions if this problem has only finitely many solutions then of these solutions there are two positive solutions two negative solutions and two sign changing solutions

amenable actions of groups the equivalence between different characterizations of amenable actions of a locally compact group is proved in particular this answers a question raised by r j zimmer in 1977

infinitely many solutions for cooperative elliptic systems with odd nonlinearity abstract by using the minimax technique in critical point theory we establish the existence of infinitely many solutions for a class of cooperative elliptic systems with odd nonlinearities our results improve several known results in the literature in contrast with the existing ones our results only demand restrictions on the behavior of the nonlinearities at infinity

sufficient conditions of standardness for filtrations of stationary processes taking values in a finite space let x be a stationary process with finite state space a bressaud et al ann probab 34 2006 1589 1600 recently provided a sufficient condition for the natural filtration of x to be standard when a has size 2 their condition involves the conditional laws p x of x0 conditionally on the whole past xk k 1 x and controls the strength of the influence of the old past of the process on its present x0 it involves the maximal gaps between p x and p y for infinite sequences x and y which coincide on their n last terms in this paper we first show that a slightly stronger result holds for any finite state space then we provide sufficient conditions for standardness based on average gaps instead of maximal gaps

on the coupled systems of second and fourth order elliptic equations in this paper by the variational method we discuss the existence and uniqueness of solutions for a coupled system of second and fourth order elliptic equations in the cases where the perturbation is zero sublinear and superlinear respectively

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

relative property t for the subequivalence relations induced by the action of sl2 z on t2 abstract let s be the equivalence relation induced by the action sl 2 z t 2 2 where 2 denotes the haar measure on the 2 torus t 2 we prove that any ergodic subequivalence relation r of s is either hyperfinite or rigid in the sense of s popa 41 the proof uses an ergodic theoretic criterion for rigidity of countable ergodic probability measure preserving equivalence relations moreover we give a purely ergodic theoretic formulation of rigidity for free ergodic probability measure preserving actions of countable groups