the number of topological generators for full groups of ergodic equivalence relations we completely elucidate the relationship between two invariants associated with an ergodic probability measure preserving pmp equivalence relation namely its cost and the minimal number of topological generators of its full group it follows that for any free pmp ergodic action of the free group on n generators the minimal number of topological generators for the full group of the action is n 1 answering a question of kechris

polish group actions dichotomies and generalized elementary embeddings the results in this paper involve two different topics in the descriptive theory of polish group actions the book becker kechris 6 is an introduction to that theory our two topics and two collections of theorems are rather unrelated but the proofs for both topics are essentially the same locally compact polish groups i e second countable locally compact groups are the traditional objects of study in the field known as topological groups more recently there has been interest in nonlocally compact polish groups such as s the symmetric group on n topologized as a subspace of nn much of this paper is concerned with a proper subclass of the class of polish groups namely those polish groups which admit a complete left invariant metric we call these cli groups the class of cli groups includes all locally compact groups proposition 3 c 2 d but also much more for example all solvable groups are cli hjorth solecki see proposition 3 c 2 f and a closed subgroup g of s is cli iff g is closed in nn see example 3 c 3 a we study continuous actions by polish groups on polish spaces and the associated orbit equivalence relation for locally compact groups all of our results are known or trivial or both but some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups one of the two topics considered in this paper is dichotomy theorems for equivalence relations these are theorems which assert that either the quotient space is small or else it contains a copy of a specific large set we prove some dichotomy theorems for the orbit space of actions by cli groups there are two types of dichotomies which we call the silver vaught dichotomy and the glimm effros dichotomy the silver vaught dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements assuming the negation of the continuum hypothesis the silver vaught dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2 0 equivalence classes the topological vaught conjecture which is open is for any continuous action by a polish group g on a polish space x the orbit equivalence relation satisfies the silver vaught dichotomy this conjecture implies and was motivated by the vaught conjecture in model theory indeed

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

incompressible viscous flows in borderline besov spaces we establish two new estimates for a transport diffusion equation as an application we treat the problem of global persistence of the besov regularity b p 1 frac 2 p 1 with p in 2 infty for the two dimensional navier stokes equations with uniform bounds on the viscosity we provide also an inviscid global result

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

orbit inequivalent actions for groups containing a copy of mathbb f _ 2 we prove that if a countable group contains a copy of mathbb f 2 then it admits uncountably many non orbit equivalent actions

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

group cocycles and the ring of affiliated operators in this article we study cocycles of discrete countable groups with values in 2 g and the ring of affiliated operators ug we clarify properties of the first cohomology of a group g with coefficients in 2 g and answer several questions from de cornulier et al transform groups 13 1 125 147 2008 moreover we obtain strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first 2 betti num ber we give numerous applications and examples of groups which satisfy our assumptions

on common fundamental domains abstract we find conditions under which two measure preserving actions of two groups on the same space have a common fundamental domain our results apply to commuting actions with separate fundamental domains lattices in groups of polynomial growth and some semidirect products we prove that two lattices of equal co volume in a group of polynomial growth one acting on the left the other on the right have a common fundamental domain

the tarski numbers of groups abstract the tarski number of a non amenable group g is the minimal number of pieces in a paradoxical decomposition of g in this paper we investigate how tarski numbers may change under various group theoretic operations using these estimates and known properties of golod shafarevich groups we show that the tarski numbers of 2 generated non amenable groups can be arbitrarily large we also use the cost of group actions to show that there exist groups with tarski numbers 5 and 6 these provide the first examples of non amenable groups without free subgroups whose tarski number has been computed precisely

existence and multiplicity of solutions for second order impulsive differential equations with dirichlet problems abstract in this paper we consider the existence and the multiplicity of solutions for second order impulsive differential equations with dirichlet boundary conditions some new existence results are obtained by critical point theory our results further improve some existing results

some unique group measure space decomposition results using an approach emerging from the theory of closable deriva tions on von neumann algebras we exhibit a class of groups cr satisfying the following property given any groups 1 2 2 cr then any free ergodic measure preserving action on a probability space 1 2 y x gives rise to a von neumann algebra with unique group measure space cartan subalgebra pairing this result with popa s orbit equivalence superrigidity theorem we obtain new examples of w superrigid actions

von neumann algebras of equivalence relations with nontrivial one cohomology abstract using popa s deformation rigidity theory we investigate prime decompositions of von neumann algebras of the form l r for countable probability measure preserving equivalence relations r we show that l r is prime whenever r is nonamenable ergodic and admits an unbounded 1 cocycle into a mixing orthogonal representation weakly contained in the regular representation this is accomplished by constructing the gaussian extension r of r and subsequently an s malleable deformation of the inclusion l r l r we go on to note a general obstruction to unique prime factorization and avoiding it we prove a unique prime factorization result for products of the form l r 1 l r 2 l r k as a corollary we get a unique factorization result in the equivalence relation setting for products of the form r 1 r 2 r k we finish with an application to the measure equivalence of groups

some oe and w rigidity results for actions by wreath product groups we use deformation rigidity theory in the von neumann algebra framework to study probability measure preserving actions by wreath product groups in particular we single out large families of wreath product groups satisfying various types of orbit equivalence oe rigidity for instance we show that whenever h k are icc property t groups such that h and k admit stably orbit equivalent action and such that h and k are ergodic then automatically is stably orbit equivalent to and h is stably orbit equivalent to k rigidity results for von neumann algebras arising from certain actions of such groups i e w rigidity results are also obtained

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

w superrigidity for bernoulli actions of property t groups we consider group measure space 1 factors m l x arising from bernoulli actions of icc property t groups more generally of groups containing an infinite normal subgroup with the relative property t and prove a rigidity result for homomorphisms m overbar m we deduce that the action x is w superrigid i e if y is any free ergodic measure preserving action such that the factors m l x and m l y are isomorphic then the actions x and y must be conjugate moreover we show that if p m 1 is a projection then pmp does not admit a group measure space decomposition nor a group von neumann algebra decomposition the latter under the additional assumption that is torsion free we also prove a rigidity result for homomorphisms m m this time for in a larger class of groups than above now including products of non amenable groups for certain groups e g f 2 f 2 we deduce that m does not embed into pmp for any projection p m 1 and obtain a description of the endomorphism semigroup of m

on the structural theory of ii1 factors of negatively curved groups ii actions by product groups abstract this paper contains a series of structural results for von neumann algebras arising from measure preserving actions by product groups on probability spaces expanding upon the methods used earlier by the first two authors we obtain new examples of strongly solid factors as well as von neumann algebras with unique or no cartan subalgebra for instance we show that every ii 1 factor associated with a weakly amenable group in the class s of ozawa is strongly solid there is also the following product version of this result any maximal abelian subalgebra of any ii 1 factor associated with a finite product of weakly amenable groups in the class s of ozawa has an amenable normalizing algebra finally pairing some of these results with a cocycle superrigidity result of ioana it follows that compact actions by finite products of lattices in s p n 1 n 2 are virtually w superrigid

compact actions whose orbit equivalence relations are not profinite abstract let x be a measure preserving action of a countable group on a standard probability space x we prove that if the action x is not profinite and satisfies a certain spectral gap condition then there does not exist a countable to one borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action i e an inverse limit of actions on countable sets as a consequence we show that if is a countable dense subgroup of a compact non profinite group g such that the left translation action g has spectral gap then g is antimodular and not orbit equivalent to any not necessarily free profinite action this provides the first such examples of compact actions partially answering a question of kechris and answering a question of tsankov

kazhdan groups have cost 1 we prove that every countably infinite group with kazhdan s property t has cost 1 answering a well known question of gaboriau it remains open if they have fixed price 1

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

new existence of homoclinic orbits for a second order hamiltonian system abstract by using the mountain pass theorem and the symmetric mountain pass theorem we establish some existence criteria to guarantee that the second order hamiltonian system u t a t u t p 2 u t w t u t 0 has at least one or infinitely many homoclinic orbits where t r u r n a c r r and w c 1 r r n r are not periodic in t our conditions on the potential w t x are rather relaxed

on the global well posedness for the axisymmetric euler equations this paper deals with the global well posedness of the 3d axisymmetric euler equations for initial data lying in critical besov spaces b p 1 1 3 p in this case the bkm criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity

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

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

operator algebraic superridigity for sl n ℤ n 3 for n 3 let sln we prove the following superridigity result for in the context of operator algebras let l be the von neumann algebra generated by the left regular representation of let m be a finite factor and let u m be its unitary group let u m be a group homomorphism such that m then either

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

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

strongly solid group factors which are not interpolated free group factors we give examples of non amenable infinite conjugacy classes groups with the haagerup property weakly amenable with constant cb 1 for which we show that the associated ii1 factors l are strongly solid i e the normalizer of any diffuse amenable subalgebra p subset l gamma generates an amenable von neumann algebra nevertheless for these examples of groups l is not isomorphic to any interpolated free group factor l ft for 1 t

l 2 betti numbers and plancherel measure we compute l 2 betti numbers of postliminal locally compact unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the plancherel measure this allows us to compute the l 2 betti numbers for semi simple lie groups with finite center simple algebraic groups over local fields and automorphism groups of locally finite trees acting transitively on the boundary c 2013 elsevier inc all rights reserved

strong rigidity of generalized bernoulli actions and computations of their symmetry groups abstract we study equivalence relations and ii 1 factors associated with quotients of generalized bernoulli actions of kazhdan groups specific families of these actions are entirely classified up to isomorphism of ii 1 factors this yields explicit computations of outer automorphism and fundamental groups in particular every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic ii 1 factors

global well posedness of helicoidal euler equations abstract this paper deals with the global existence and uniqueness results for the three dimensional incompressible euler equations with a particular structure for initial data lying in critical spaces in this case the bkm criterion is not known

stabilizers for ergodic actions of higher rank semisimple groups let g be a connected semisimple lie group with finite center and r rank 2 suppose that each simple factor of g either has r rank 2 or is locally isomorphic to sp l n or f4 20 we prove that any faithful irreducible properly ergodic finite measure preserving action of g is essentially free we extend the result to reducible actions and actions of lattices

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

fast homoclinic solutions for a class of damped vibration problems in this paper we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu t q t u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t our approach is variational and it is based on the critical point theory we prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function moreover some open problems proposed by zhang and ruan are resolved our theorems appear to be the first such result

unique cartan decomposition for ii1 factors arising from arbitrary actions of free groups we prove that for any free ergodic probability measure preserving action mathbb f n curvearrowright x mu of a free group on n generators mathbb f n 2 leq n leq infty the associated group measure space ii1 factor l infty x rtimes mathbb f n has l x as its unique cartan subalgebra up to unitary conjugacy we deduce that group measure space ii1 factors arising from actions of free groups with different number of generators are never isomorphic we actually prove unique cartan decomposition results for ii1 factors arising from arbitrary actions of a much larger family of groups including all free products of amenable groups and their direct products

hnn extensions of von neumann algebras abstract reduced hnn extensions of von neumann algebras as well as c algebras will be introduced and their modular theory factoriality and ultraproducts will be discussed in several concrete settings detailed analysis on them will be also carried out

free fisher information with respect to a completely positive map and cost of equivalence relations given a family of isometries in a tracial von neumann algebra m a unital subalgebra b m and a completely positive map we define the free fisher information of relative to b and using this notion we define the free dimension of relative to b id

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

stabilizers of ergodic actions of lattices and commensurators we prove that any ergodic measure preserving action of an irreducible lattice in a semisimple group with finite center and each simple factor having rank at least two either has finite orbits or has finite stabilizers the same dichotomy holds for many commensurators of such lattices the above are derived from more general results on groups with the howe moore property and property t we prove similar results for commensurators in such groups and for irreducible lattices and commensurators in products of at least two such groups at least one of which is totally disconnected

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

actions of _ whose ii factors and orbit equivalence relations have prescribed fundamental group we show that given any subgroup f of r which is either countable or belongs to a certain large class of uncountable subgroups there exist continuously many free ergodic measure preserving actions i of the free group with infinitely many generators f1 on probability measure spaces xi i such that their associated group measure space ii1 factors mi l 1 xi o i f1 and orbit equivalence relationsri r f1 y xi have fundamental group equal tof and with mi respectivelyri stably non isomorphic moreover these actions can be taken so thatri has no outer automorphisms and any automorphism of mi is unitary conjugate to an automorphism that acts trivially on the subalgebra l 1 xi of mi

l 2 betti numbers of locally compact groups and their cross section equivalence relations we prove that the l betti numbers of a unimodular locally compact group g coincide up to a natural scaling constant with the l betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of g as a consequence we obtain that the reduced and un reduced l betti numbers of g agree and that the l betti numbers of a lattice in g equal those of g up to scaling by the covolume of in g we also deduce several vanishing results including the vanishing of the reduced l cohomology for amenable locally compact groups

rigidity for von neumann algebras given by locally compact groups and their crossed products we prove the first rigidity and classification theorems for crossed product von neumann algebras given by actions of non discrete locally compact groups we prove that for arbitrary free probability measure preserving actions of connected simple lie groups of real rank one the crossed product has a unique cartan subalgebra up to unitary conjugacy we then deduce a w strong rigidity theorem for irreducible actions of products of such groups more generally our results hold for products of locally compact groups that are nonamenable weakly amenable and that belong to ozawa s class mathcal s

measure equivalence rigidity and bi exactness of groups we get three types of results on measurable group theory direct product groups of ozawa s class s groups wreath product groups and amalgamated free products we prove measure equivalence factorization results on direct product groups of ozawa s class s groups as consequences monod shalom type orbit equivalence rigidity theorems follow we prove that if two wreath product groups a g b of non amenable exact direct product groups g with amenable bases a b are measure equivalent then g and are measure equivalent we get bass serre rigidity results on amalgamated free products of non amenable exact direct product groups

measurable quotients of unipotent translations on homogeneous spaces let u be a nilpotent unipotent subgroup of a lie group g and let f be a closed subgroup of g marina ratner showed that every ergodic u invariant probability measure on the homogeneous space f g is of a simple algebraic form we use this fundamental new result to show that every measurable quotient of the u action on j g is of a simple algebraic form roughly speaking any quotient is a double coset space a g k

continuous cocycle superrigidity for shifts and groups with one end we prove that if a finitely generated group g is not torsion then a necessary and sufficient condition for every full shift over g has continuous cocycle superrigidity is that g has one end it is a topological version of the well known popa s measurable cocycle superrigidity theorem cocycle and orbit equivalence superrigidity for malleable actions of w rigid groups invent math 170 2 243 295 2007 for the proof of sufficiency we introduce a new specification property for shifts over general groups which plays a similar role as malleable property in the measurable setting this new specification property is good enough for us to extend the method of using homoclinic equivalence relation that was introduced by schmidt pac j math 170 1 237 269 1995 to study cocycle rigidity for mathbb z d shifts indeed in this direction we prove this superrigidity result for certain more general systems and for the necessary condition we apply specker s characterization for ends of groups via the associated first cohomology groups to get the result finally combining our results with li ergod theory dyn syst arxiv 1503 01704 theorem 1 6 we have an application in continuous orbit equivalence rigidity

on the superrigidity of malleable actions with spectral gap some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces v rx x are revealed by the study of group measure space von neumann algebras l x x t mvnl and the classification of actions up to orbit equivalence oe i e up to isomorphism of probability spaces carrying the orbits of actions onto each other although one is in von neumann algebras and the other in ergodic theory the two problems are closely related as an oe of actions v rx x a rxy has been shown to implement an algebra isomorphism l x x t l y x a taking l x onto l y and vice versa si dy fm in particular the isomorphism class of l x xi t only depends on the equivalence relation 7 r t gt 1t x g t thus von neumann equivalence vne of group actions requiring isomorphism of their group measure space algebras is weaker than oe since there are exam ples of non oe actions whose associated von neumann algebras are all isomorphic cj1 it is in general strictly weaker on the other hand oe is manifestly weaker than classical conjugacy which for free actions r rx x a rx y requires isomor phism of probability spaces a x p y v satisfying ar a 1 a so in particular r a how much weaker vne and oe can be with respect to conju gacy is best seen in the amenable case where by a celebrated theorem of connes all free ergodic actions of all infinite amenable groups give rise to the same hi factor cl and by dy ow cfw they are undistinguishable under oe as well also any embedding of algebras l x xi t c l y x a with a amenable forces t to be amenable but the non amenable case is extremely complex and for many years progress has been slow mvn2 dy me c2 cw sc even after the discovery of the first rigidity phenomena by connes in von neumann algebras c3 c4 and by zimmer in oe ergodic theory zi z2 this changed dramatically over the last 7 8 years with the advent of a variety of striking rigidity results ful gl g2 mosh h hk p1 p8 see p9 for a survey also sh for a survey on oe rigidity our aim in this paper is to investigate the most extreme such phenomena called strong rigidity which show that for certain classes of source group actions

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

cocycle superrigidity for translation actions of product groups let g be either a profinite or a connected compact group and be finitely generated dense subgroups assuming that the left translation action of on g is strongly ergodic we prove that any cocycle for the left right translation action of on g with values in a countable group is virtually cohomologous to a group homomorphism moreover we prove that the same holds if g is a not necessarily compact connected simple lie group provided that contains an infinite cyclic subgroup with compact closure we derive several applications to oe and w superrigidity in particular we obtain the first examples of compact actions of f2 f2 which are w superrigid

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

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

on a measurable analogue of small topological full groups abstract we initiate the study of a measurable analogue of small topological full groups that we call l 1 full groups these groups are endowed with a polish group topology which admits a natural complete right invariant metric we mostly focus on l 1 full groups of measure preserving z actions which are actually a complete invariant of flip conjugacy we prove that for ergodic actions the closure of the derived group is topologically simple although it can fail to be simple we also show that the closure of the derived group is connected and that for measure preserving free actions of non amenable groups the closure of the derived group and the l 1 full group itself are never amenable in the case of a measure preserving ergodic z action the closure of the derived group is shown to be the kernel of the index map if such an action is moreover by homeomorphism on the cantor space we show that the topological full group is dense in the l 1 full group using juschenko monod and matui s results on topological full groups we conclude that l 1 full groups of ergodic z actions are amenable as topological groups and that they are topologically finitely generated if and only if the z action has finite entropy

incompressible flows of an ideal fluid with unbounded vorticity in this paper we study solutions to the euler equations of an ideal incompressible fluid in rn singular at the origin with a finite symmetry group for an admissible class of finite groups we prove a local existence and uniqueness theorem in even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity in arbitrary dimension including n 3 we construct local in time solutions with vorticity that behaves e g like a function of homogeneous degree zero near the origin the symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

the norm of the euler class we prove that the norm of the euler class mathcal e for flat vector bundles is 2 n in even dimension n since it vanishes in odd dimension this shows that the sullivan smillie bound considered by gromov and ivanov turaev is sharp in the course of the proof we construct a new cocycle representing mathcal e and taking only the two values 2 n furthermore we establish the uniqueness of a canonical bounded euler class

orbit equivalence rigidity consider a countable group acting ergodically by measure p reserving transformations on a probability space x and let r be the corresponding orbit equivalence relation on x the following rigidity phenomenon is shown there exist group actions such that the equivalence relation r on x determines the group and the action x uniquely up to finite groups the natural action of sln z on the n torus r n z n for n 2 is one of such examples the interpretation of these results in the context of von neumann algebras provides some support to the conjecture of connes on rigidity of group algebras for groups with property t our rigidity results also give examples of countable equivalence relations of type ii1 which cannot be generated mod 0 by a free action of any group this gives a negative answer to a long standing problem of feldman and moore