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

a generalisation to birkhoff von neumann theorem abstract the classic birkhoff von neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices in this paper we study a generalisation of this theorem in the type ii 1 setting namely we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms of the unit interval with similar properties we show that a weaker version of this theorem still holds

a measurable group theoretic solution to von neumann s problem we give a positive answer in the measurable group theory context to von neumann s problem of knowing whether a non amenable countable discrete group contains a non cyclic free subgroup we also get an embedding result of the free group von neumann factor into restricted wreath product factors

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

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

a theory of period doubling bifurcations in two dimensional reversible area preserving mappings for a class of two dimensional reversible area preserving mappings a theory is developed which describes how the period doubling bifurcation entails the onset of hyperbolic instability with reflection henon s quadratic mapping chirikov s standard mapping and de vogelaere s mapping all belong to this class

on the stable analytic continuation with a condition of uniform boundedness it is shown that if h x is any continuous function defined on some interval a b 1 1 of the real axis then in general its best l2 approximant in the class of functions holomorphic and bounded by unity in the unit disk of the complex plane is a finite blaschke product an upper bound is placed on the number of factors of the latter and a method for its construction is given the paper contains a discussion of the use of these results in performing a stable analytic continuation of a set of data points under a condition of uniform boundedness as well as some numerical examples

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

rank combinatorial cost and homology torsion growth in higher rank lattices we investigate the rank gradient and growth of torsion in homology in residually finite groups as a tool we introduce a new complexity notion for generating sets using measured groupoids and combinatorial cost as an application we prove the vanishing of the above invariants for farber sequences of subgroups of right angled groups a group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute most nonuniform lattices in higher rank simple lie groups are right angled we provide the first examples of uniform cocompact right angled arithmetic groups in sl n n 3 and so p q for some values of p q this is a class of lattices for which the congruence subgroup property is not known in general by using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple lie group 2017

on sofic actions and equivalence relations abstract the notion of sofic equivalence relation was introduced by gabor elek and gabor lippner their technics employ some graph theory here we define this notion in a more operator algebraic context starting from connes embedding problem and prove the equivalence of these two definitions we introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic we also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic

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

tupling in three dimensional reversible mappings feigenbaum s parameter scaling exponents s are calculated for l 3m l 5m l 7m l 9m sequences of periodic orbits in a class of three dimensional non measure preserving reversible mappings some preliminary results on orbit scaling exponents for l 3m sequences are also included scaling exponents found so far are the same as those found for two dimensional reversible mappings

a converse to dye s theorem every non amenable countable group induces orbit inequivalent ergodic equivalence relations on standard borel probability spaces not every free ergodic measure preserving action of f 2 on a standard borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces there is a treeable non hyperfinite borel equivalence relation which is not universal for treeable in the b ordering

the universality of period doubling bifurcations in certain two dimensional reversible area preserving mappings with quadratic nonlinearity the author shows that all the period doubling bifurcation sequences of two dimensional one parameter reversible area preserving mappings of a certain form with quadratic nonlinearity are equivalent two such sequences are calculated according to the theory developed by the author lee and chai 1983 for the explicit demonstration of the equivalence

stability for product groups and property τ abstract we study the notion of permutation stability or p stability for countable groups our main result provides a wide class of non amenable product groups which are not p stable this class includes the product group whenever admits a non abelian free quotient and admits an infinite cyclic quotient in particular we obtain that the groups f m z d and f m f n are not p stable for any integers m n 2 and d 1 this implies that p stability is not closed under the direct product construction which answers a question of becker lubotzky and thom the proof of our main result relies on a construction of asymptotic homomorphisms from to finite symmetric groups starting from sequences of finite index subgroups in and with and without property our method is sufficiently robust to show that the groups covered are not even flexibly p stable thus giving the first such non amenable residually finite examples

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

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

free products of measured equivalence relations abstract for given countable standard equivalence relations over the same unit space we construct their free product which is a measured groupoid related topics such as measures and 2 cocyles on the free product needed to deal with groupoid von neumann algebras are also discussed

modular actions and amenable representations consider a measure preserving action x of a countable group and a measurable cocycle x aut y with countable image where x is a standard lebesgue space and y is any probability space we prove that if the koopman representation associated to the action x is non amenable then there does not exist a countable to one borel homomorphism from the orbit equivalence relation of the skew product action x y to the orbit equivalence relation of any modular action i e an inverse limit of actions on countable sets or equivalently an action on the boundary of a countably splitting tree generalizing previous results of hjorth and kechris as an application for certain groups we connect antimodularity to mixing conditions we also show that any countable non amenable residually finite group induces at least three mutually orbit inequivalent free measure preserving ergodic actions as well as two non borel bireducible ones

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

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

normal generation and ell 2 betti numbers of groups the normal rank of a group is the minimal number of elements whose normal closure coincides with the group we study the relation between the normal rank of a group and its first ell 2 betti number and conjecture the inequality beta 1 2 g le mathrm nrk g 1 for torsion free groups the conjecture is proved for limits of left orderable amenable groups on the other hand for every n ge 2 and every varepsilon 0 we give an example of a simple group q with torsion such that beta 1 2 q ge n 1 varepsilon these groups also provide examples of simple groups of rank exactly n for every n ge 2 existence of such examples for n 3 was unknown until now

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

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

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

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

minimal spanning forests minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs these limits can be taken with free or wired boundary conditions and are denoted fmsf free minimal spanning forest and wmsf wired minimal spanning forest respectively the wmsf is also the union of the trees that arise from invasion percolation started at all vertices we show that on any cayley graph where critical percolation has no infinite clusters all the component trees in the wmsf have one end a s in z d this was proved by alexander ann probab 23 1995 87 104 but a different method is needed for the nonamenable case we also prove that the wmsf components are thin in a different sense namely on any graph each component tree in the wmsf has p c 1 a s where p c denotes the critical probability for having an infinite cluster in bernoulli percolation on the other hand the fmsf is shown to be thick on any connected graph the union of the fmsf and independent bernoulli percolation with arbitrarily small parameter is a s connected in conjunction with a recent result of gaboriau this implies that in any cayley graph the expected degree of the fmsf is at least the expected degree of the fsf the weak limit of uniform spanning trees we also show that the number of infinite clusters for bernoulli p u percolation is at most the number of components of the fmsf where p u denotes the critical probability for having a unique infinite cluster finally an example is given to show that the minimal spanning tree measure does not have negative associations

a class of ii _1 factors with a unique mcduff decomposition we provide a fairly large class of ii 1 factors n such that m n bar otimes r has a unique mcduff decomposition up to isomorphism where r denotes the hyperfinite ii 1 factor this class includes all ii 1 factors n l infty x rtimes gamma associated to free ergodic probability measure preserving p m p actions gamma curvearrowright x mu such that either a gamma is a free group mathbb f n for some n ge 2 or b gamma is a non inner amenable group and the orbit equivalence relation of the action gamma curvearrowright x mu satisfies a property introduced in jones and schmidt am j math 109 1 91 114 1987 on the other hand settling a problem posed by jones and schmidt in 1985 we give the first examples of countable ergodic p m p equivalence relations which do not satisfy the property of jones and schmidt 1987 we also prove that if mathcal r is a countable strongly ergodic p m p equivalence relation and mathcal t is a hyperfinite ergodic p m p equivalence relation then mathcal r times mathcal t has a unique stable decomposition up to isomorphism finally we provide new characterisations of property gamma for ii 1 factors and of strong ergodicity for countable p m p equivalence relations

analytic continuation from data points with unequal errors the problem solved in this paper is that of constructing zero free holomorphic functions which will a assume specified values at a set of discrete data points in a data region 1 inside the holomorphy domain and which at the same time will b provide the optimum solution to various stabilizing boundedness or smoothness conditions on the boundaries r the immediate motivation to this problem arose from the need to renormalize data with unequal errors by using a holomorphic weight function to bring all the errors to the same value this was a preliminary step before making an analytic continuation off the data region 1 since a stabilizing condition has to be imposed on the boundaries r the weight function must be chosen so as to introduce the minimum additional instability on r although this was the specific motivation other interesting applications suggest themselves and some of these are discussed the stability conditions on r which are treated may all be expressed in terms of the real par

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

optimal analytic extrapolations revisited the problem of optimal analytical extrapolation of holomorphic functions from a finite set of interior data points to another interior point is completely solved in the general case of data known with unequal errors simple and easy to handle algorithms are obtained

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

free product actions with relative property t and trivial outer automorphism groups abstract we show that every non amenable free product of groups admits free ergodic probability measure preserving actions which have relative property t in the sense of s popa 2006 pop06 def 4 1 there are continuum many such actions up to orbit equivalence and von neumann equivalence and they may be chosen to be conjugate to any prescribed action when restricted to the free factors we exhibit also for every non amenable free product of groups free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group this gives in particular the first examples of such actions for the free group on 2 generators

generator problem for certain property t factors we show that the property t factors associated with groups sln z n 3 are generated by two selfadjoint elements one of which has an arbitrarily small support this answers a question of dan voiculescu

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

period doubling bifurcations and universality in conservative systems we have studied numerically a sequence of period doubling bifurcations of periodic orbits of henon s conservative two dimensional mapping in agreement with other recent studies we also find evidence that such sequences possess universality properties similar to the ones observed for dissipative systems the corresponding universal constants however have significantly different values in each case we suggest that a possible explanation for this discrepancy may be the fact that conservative systems owing to their measure preserving property do not become asymptotically one dimensional as dissipative systems do

volume gradients and homology in towers of residually free groups we study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups g n g of increasing finite index in a fixed finitely generated group g assuming bigcap n g n 1 we focus in particular on finitely presented residually free groups calculating their ell 2 betti numbers rank gradient and asymptotic deficiency if g is a limit group and k is any field then for all j ge 1 the limit of dim h j g n k g g n as n rightarrow infty exists and is zero except for j 1 where it equals chi g we prove a homotopical version of this theorem in which the dimension of dim h j g n k is replaced by the minimal number of j cells in a k g n 1 this includes a calculation of the rank gradient and the asymptotic deficiency of g both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of slow groups we prove that if a residually free group g is of type mathrm fp m but not of type mathrm fp infty then there exists an exhausting filtration by normal subgroups of finite index g n so that lim n dim h j g n k g g n 0 hbox for j le m if g is of type mathrm fp infty then the limit exists in all dimensions and we calculate it

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

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

hyperfinite and ii1 actions for nonamenable groups abstract the actions of certain nonamenable groups on the lebesgue space are studied an example is constructed of a group which has a continuum of weakly nonequivalent actions of type ii 1 it is also proved that if a free group with two generators has hyperfinite action on the lebesgue space then at least one generator acts dissipatively a hyperfinite action is constructed for any nonamenable group

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