kazhdan projections random walks and ergodic theorems in this paper we investigate generalizations of kazhdan s property t to the setting of uniformly convex banach spaces we explain the interplay between the existence of spectral gaps and that of kazhdan projections our methods employ markov operators associated to a random walk on the group for which we provide new norm estimates and convergence results this construction exhibits useful properties and flexibility and allows to reclassify the existence of kazhdan projections from a c algebraic phenomenon to a natural object associated to random walks on groups we give a number of applications of these results in particular we address several open questions we give a direct comparison of properties te and fe with lafforgue s reinforced banach property t we obtain shrinking target theorems for orbits of kazhdan groups and apply them to a question of kleinbock and margulis finally we construct non compact ghost projections for warped cones answering a question of willett and yu in this last case we conjecture that such warped cones provide new counterexamples to the coarse baum connes conjecture

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

quasi shadowing for partially hyperbolic dynamics on banach spaces abstract a partially hyperbolic dynamical system is said to have the quasi shadowing property if every pseudotrajectory can be shadowed by a sequence of points x n n z such that x n 1 is obtained from the image of x n by moving it by a small factor in the central direction in the present paper we prove that a small nonlinear perturbation of a partially dichotomic sequence of not necessarily invertible linear operators acting on an arbitrary banach space has the quasi shadowing property we also obtain a continuous time version of this result as an application of our main result we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction

on ii1 factors arising from 2 cocycles of w rigid groups we consider ii1 factors l g arising from 2 cocyles h2 g t on groups g containing infinite normal subgroups h g with the relative property t i e g w rigid we prove that given any separable ii1 factor m the set of 2 cocycles h h2 h t with the property that l g is embeddable into m is at most countable we use this result the relative property t of z2 z2 for sl 2 z non amenable and the fact that every cocycle h2 z2 t t extends to a cocycle on z2 sl 2 z to show that the one parameter family of ii1 factors m l z2 t are mutually non isomorphic modulo countable sets and cannot all be embedded into the same separable ii1 factor other examples and applications are discussed

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

some prime factorization results for type ii1 factors we prove several unique prime factorization results for tensor products of type ii1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected lie groups of real rank 1 in particular we show that if mathcal r overline otimes mathcal l mathbb f r 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f r m is isomorphic to a subfactor in mathcal r overline otimes mathcal l mathbb f s 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f s n for some 2 ri sj then m n

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

sharp regularity of linearization for c 1 1 hyperbolic diffeomorphisms c 1 linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions however c 1 smoothness is not enough to guarantee c 1 linearization for c 1 1 hyperbolic diffeomorphisms on banach spaces c 1 linearization was proved under a gap condition together with a band condition of the spectrum in this paper the result of c 1 linearization in banach spaces is strengthened to c 1 beta linearization with a constant beta 0 under a weaker band condition by a decomposition with invariant foliations the weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum furthermore we give an estimate for the exponent beta and prove that the estimate is sharp in the planar case

kazhdan constants of some colored permutation groups abstract colored permutation groups serve as a generalization of coxeter groups of types a and b we compute exact kazhdan constants for the coxeter groups of types b and d and for colored permutation groups the computations rely on the combinatorics of young tableaux and on suitable decompositions of representation spaces

property t for noncommutative universal lattices we establish a new spectral criterion for kazhdan s property t which is applicable to a large class of discrete groups defined by generators and relations as the main application we prove property t for the groups el n r where n 3 and r is an arbitrary finitely generated associative ring we also strengthen some of the results on property t for kac moody groups from dymara and januszkiewicz in invent math 150 3 579 627 2002

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

local contractions of banach spaces and spectral gap conditions abstract we study the linearization problem for a c k 1 k 1 contraction of a banach space e near a fixed point which satisfies a spectral gap condition and a narrow band condition both of order k we also assume that the part of the spectrum in each band satisfies a finite non resonant condition of order k relative to itself together with the part that lies in the larger bands we show that there is a c k linearization for sufficiently small 0 we give a precise estimate on in terms of the gap and band conditions

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

entropy and distance of random graphs with application to structural pattern recognition the notion of a random graph is formally defined it deals with both the probabilistic and the structural aspects of relational data by interpreting an ensemble of attributed graphs as the outcomes of a random graph we can use its lower order distribution to characterize the ensemble to reflect the variability of a random graph shannon s entropy measure is used to synthesize an ensemble of attributed graphs into the distribution of a random graph or a set of distributions we propose a distance measure between random graphs based on the minimum change of entropy before and after their merging when the ensemble contains more than one class of pattern graphs the synthesis process yields distributions corresponding to various classes this process corresponds to unsupervised learning in pattern classification using the maximum likelihood rule and the probability computed for the pattern graph based on its matching with the random graph distributions of different classes we can classify the pattern graph to a class

structural matching by discrete relaxation this paper describes a bayesian framework for performing relational graph matching by discrete relaxation our basic aim is to draw on this framework to provide a comparative evaluation of a number of contrasting approaches to relational matching broadly speaking there are two main aspects to this study firstly we focus on the issue of how relational inexactness may be quantified we illustrate that several popular relational distance measures can be recovered as specific limiting cases of the bayesian consistency measure the second aspect of our comparison concerns the way in which structural inexactness is controlled we investigate three different realizations of the matching process which draw on contrasting control models the main conclusion of our study is that the active process of graph editing outperforms the alternatives in terms of its ability to effectively control a large population of contaminating clutter

poincare type theorems for non autonomous systems in this paper we establish analytic equivalence theorems of poincare and poincare dulac type for analytic non autonomous differential systems based on the dichotomy spectrum of their linear part as applications of the theorem normal forms linearize for two illustrative examples

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

weak exactness for c algebras and application to condition ao abstract kirchberg introduced weak exactness for von neumann algebras as an analogue of exactness for c algebras ozawa found useful characterizations of weak exactness and he proved that weak exactness for group von neumann algebras is equivalent to exactness of the groups we generalize this concept to inclusions of c algebras in von neumann algebras and study some fundamental properties and relationship to kirchberg s weak exactness we then give some permanence properties which are similar to those of exact groups in the last section we study a similar condition to ozawa s condition ao with our weak exactness and generalize ozawa s theorem for bi exact groups as a corollary we give new examples of prime factors

function described graphs for modelling objects represented by sets of attributed graphs we present in this article the model function described graph fdg which is a type of compact representation of a set of attributed graphs ags that borrow from random graphs the capability of probabilistic modelling of structural and attribute information we define the fdgs their features and two distance measures between ags unclassified patterns and fdgs models or classes and we also explain an efficient matching algorithm two applications of fdgs are presented in the former fdgs are used for modelling and matching 3d objects described by multiple views whereas in the latter they are used for representing and recognising human faces described also by several views

smooth linearization of nonautonomous difference equations with a nonuniform dichotomy there have been obtained many results on smooth linearization for diffeomorphisms those results cannot be simply applied to nonautonomous difference equations in this paper we investigate c 1 smooth linearization for nonautonomous difference equations with a nonuniform strong exponential dichotomy reducing the linear part of the nonautonomous system defined by a sequence of invertible linear operators on mathbb r d to a bounded linear operator on a banach space we discuss the spectrum and its spectral gaps then we obtain a gap condition for c 1 linearization of such a nonautonomous difference equation we finally extend the result to the infinite dimensional case our theorems improve known results even in the case of uniform strong exponential dichotomies

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

unsupervised contour closure algorithm for range image edge based segmentation this paper presents an efficient technique for extracting closed contours from range images edge points edge points are assumed to be given as input to the algorithm i e previously computed by an edge based range image segmentation technique the proposed approach consists of three steps initially a partially connected graph is generated from those input points then the minimum spanning tree of that graph is computed finally a postprocessing technique generates a single path through the regions boundaries by removing noisy links and closing open contours the novelty of the proposed approach lies in the fact that by representing edge points as nodes of a partially connected graph it reduces the contour closure problem to a minimum spanning tree partitioning problem plus a cost function minimization stage to generate closed contours experimental results with synthetic and real range images together with comparisons with a previous technique are presented

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

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

differentiability with respect to parameters in global smooth linearization abstract let x be a banach space and t x x a family of invertible contractions t l f where l is linear and f is nonlinear with f 0 0 we give conditions for the existence of a family of global linearization maps h such that h t h 1 l with a smooth dependence on the results depend strongly on the choice of some appropriate spaces of maps adapted norms and the use of a specific fixed point theorem with smooth dependence on parameters

linearization of class c1 for contractions on banach spaces in this work we prove a c 1 linearization result for contraction diffeomorphisms near a fixed point valid in infinite dimensional banach spaces as an intermediate step we prove a specific result of existence of invariant manifolds which can be interesting by itself and that was needed on the proof of our main theorem our results essentially generalize some classical results by p hartman in finite dimensions and a result of x mora j sola morales in the infinite dimensional case it is shown that the result can be applied to some abstract systems of semilinear damped wave equations

universal lattices and unbounded rank expanders we study the representations of non commutative universal lattices and use them to compute lower bounds of the constant for the commutative universal lattices g d k sl d x 1 x k for d 3 with respect to several generating sets as an application we show that the cayley graphs of the finite groups text sl 3k mathbb f p can be made expanders with a suitable choice of generators this provides the first example of expander families of groups of lie type where the rank is not bounded and provides counter examples to two conjectures of a lubotzky and b weiss

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

on relative property t and haagerup s property abstract consider the following three properties for countable discrete groups g 1 g has an infinite subgroup with relative property t 2 the group von neumann algebra l g has a diffuse von neumann subalgebra with relative property t and 3 g does not have haagerup s property it is clear that 1 implies 2 and 2 implies 3 i will prove that both of the converses are false this is joint work with ionut chifan

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

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

an optimal graph theoretic approach to data clustering theory and its application to image segmentation a novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated the data to be clustered are represented by an undirected adjacency graph g with arc capacities assigned to reflect the similarity between the linked vertices clustering is achieved by removing arcs of g to form mutually exclusive subgraphs such that the largest inter subgraph maximum flow is minimized for graphs of moderate size approximately 2000 vertices the optimal solution is obtained through partitioning a flow and cut equivalent tree of g which can be efficiently constructed using the gomory hu algorithm 1961 however for larger graphs this approach is impractical new theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size this algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices the new clustering algorithm is applied to the image segmentation problem the segmentation is achieved by effectively searching for closed contours of edge elements equivalent to minimum cuts in g which consist mostly of strong edges while rejecting contours containing isolated strong edges this method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours

filtering graphs to check isomorphism and extracting mapping by using the conductance electrical model this paper presents a new method of filtering graphs to check exact graph isomorphism and extracting their mapping each graph is modeled by a resistive electrical circuit using the conductance electrical model cem by using this model a necessary condition to check the isomorphism of two graphs is that their equivalent resistances have the same values but this is not enough and we have to look for their mapping to find the sufficient condition we can compute the isomorphism between two graphs in o n 3 where n is the order of the graph if their star resistance values are different otherwise the computational time is exponential but only with respect to the number of repeated star resistance values which usually is very small we can use this technique to filter graphs that are not isomorphic and in case that they are we can obtain their node mapping a distinguishing feature over other methods is that even if there exists repeated star resistance values we can extract a partial node mapping of all the nodes except the repeated ones and their neighbors in o n 3 the paper presents the method and its application to detect isomorphic graphs in two well know graph databases where some graphs have more than 600 nodes highlightsthe method uses the cem to model the graphs and detect the isomorphic graphs the time complexity for detecting non isomorphic graphs of n degree is in most of the cases o n3 the cem can obtain the complete or partial node mapping between the graphs the time complexity for detecting graphs isomorphism in the worst time complexity can be exponential with respect to the number of repeated star values the detection and mapping of a graph isomorphism excluding co resistances are clearly separated which allows doing the filter process in two phases the filter performs an early detection of non isomorphic graphs the filtering process is not probabilistic not iterative neither recursive so the computation complexity is deterministic and only depends on n we do not need to calculate the pseudoinverse we can compute the star resistance values by a sum of finite number of terms

the product replacement algorithm and kazhdan s property t a problem of great importance in computational group theory is to generate nearly uniformly distributed random elements in a finite groupg a good example of such an algorithm should start at any given set of generators use no prior knowledge of the structure of g and in a polynomial number of group operations return the desired random group elements see bb2 then these random elements can be used further to determine the structure of g in a pioneer paper bb1 babai found such an algorithm which provably generates nearly uniformly distributed random elements in o log g group multiplications too many for practical applications a different heuristic the product replacement algorithm was designed by leedham green and soicher lg and studied by celler et al in clmno in spite of the fact that very little theoretical justification was known practical experiments showed excellent performance so it quickly became the most popular practical algorithm to generate random group elements and was included in the two most frequently used group algebra packages gap sc and magma bcp a systematic and quantitative approach was carried out by diaconis and saloffcoste ds1 ds2 see also bb2 cg but their results did not reveal the mystery of the truly outstanding performance of the algorithm the aim of this paper is to propose a conceptual explanation based on kazhdan s property t from representation theory of lie groups and to improve some of the previous estimates on the running time the product replacement algorithm works as follows clmno given a finite group g let k g be the set of k tuples g g1 gk of elements of g such that g1 gk g we call elements of k g the generating k tuples given a generating k tuple g1 gk define a move on it in the following way choose uniformly a pair i j such that 1 i 6 j k then apply one of the following four operations with equal probability

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

c1 linearization for planar contractions c1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems it is known that planar c1 contractions with a fixed point at the origin o admit c1 linearization with sufficiently small 0 if 1 and admit c1 linearization if log 1 log 2 1 1 where 1 and 2 are eigenvalues of the linear parts of the contractions at o with 0 1 2 1 in this paper we improve the lower bound of to lower the condition of c1 linearization for planar contractions furthermore we prove that the derivatives of transformations in our c1 linearization are holder continuous and give estimates for the holder exponent finally we give a counter example to show that those estimates cannot be improved anymore

uniform non amenability abstract for any finitely generated group g an invariant fol g 0 is introduced which measures the amount of non amenability of g if g is amenable then fol g 0 if fol g 0 we call g uniformly non amenable we study the basic properties of this invariant for example its behaviour when passing to subgroups and quotients of g we prove that the following classes of groups are uniformly non amenable non abelian free groups non elementary word hyperbolic groups large groups free burnside groups of large enough odd exponent and groups acting acylindrically on a tree uniform non amenability implies uniform exponential growth we also exhibit a family of non amenable groups in particular including all non solvable baumslag solitar groups which are not uniformly non amenable that is they satisfy fol g 0 finally we derive a relation between our uniform folner constant and the uniform kazhdan constant with respect to the left regular representation of g

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

cocycle superrigidity and bounded cohomology for negatively curved spaces we introduce new techniques to extend superrigidity theory beyond the scope of lie or algebraic groups we construct a cohomological invariant which accounts for and generalizes all known superrigidity results for actions on negatively curved spaces together with a new vanishing result and the machinery of bounded cohomology this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature we also prove a cocycle version a la zimmer

measure reducibility of countable borel equivalence relations we show that every basis for the countable borel equivalence relations strictly above mathbb e 0 under measure reducibility is uncountable thereby ruling out natural generalizations of the glimm effros dichotomy we also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base using arguments substantially simpler than those previously employed

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

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

uniform pointwise bounds for matrix coefficients of unitary representations and applications to kazhdan constants let k be a local field and g the group of k rational points of a connected reductive linear algebraic group over k with k semisimple rank g 2 let k be a good maximal compact subgroup of g for a unitary representation of g a vector v in is called kfinite if the subspace spanned bykv is finite dimensional we will use the termk matrix coefficients resp k finite matrix coefficients of to refer to its matrix coefficients with respect to k invariant resp k finite unit vectors following bt we denote by g the subgroup generated by the unipotent k split subgroups of g the main goal of the present paper is to construct a class of uniform pointwise bounds for the k finite matrix coefficients of all infinite dimensional irreducible unitary representations of g or more generally of all unitary representations of g without a non zero g invariant vector let a be a maximal k split torus and a the closed positive weyl chamber of a such that the cartan decomposition g ka k holds where is a finite subset of the

actions of rigid groups on uhf algebras let be a countably infinite property t group and let d be uhf algebra of infinite type we prove that there exists a continuum of pairwise non weakly cocycle conjugate strongly outer actions of on d the proof consists in assigning to any second countable abelian pro p group g a strongly outer action of on d whose weak cocycle conjugacy class completely remembers the group g the group g is reconstructed from the action through its weak 1 cohomology set endowed with a canonical pairing function our construction also shows the following stronger statement the relations of conjugacy cocycle conjugacy and weak cocycle conjugacy of strongly outer actions of on d are complete analytic sets and in particular not borel the same conclusions hold more generally when is only assumed to contain an infinite subgroup with relative property t and for actions on not necessarily simple separable nuclear uhf absorbing self absorbing c algebras with at least one trace finally we use the techniques of this paper to construct outer actions on r with prescribed cohomology precisely for every infinite property t group and for every countable abelian group we construct an outer action of on r whose 1 cohomology is isomorphic to

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

reduced 1 cohomology and relative property t shalom characterized property t in terms of the vanishing of all reduced first cohomology for compactly generated groups we characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes we show that in a strong sense this is inequivalent to relative property t even for compactly generated group pairs

graph based representations and techniques for image processing and image analysis in this paper we will discuss the use of some graph based representations and techniques for image processing and analysis instead of making an extensive review of the graph techniques in this field we will explain how we are using these techniques in an active vision system for an autonomous mobile robot developed in the institut de robotica i informatica industrial within the project active vision system with automatic learning capacity for industrial applications cicyt tap98 0473 specifically we will discuss the use of graph based representations and techniques for image segmentation image perceptual grouping and object recognition we first present a generalisation of a graph partitioning greedy algorithm for colour image segmentation next we describe a novel fusion of colour based segmentation and depth from stereo that yields a graph representing every object in the scene finally we describe a new representation of a set of attributed graphs ags denominated function described graphs fdgs a distance measure for matching ags with fdgs and some applications for robot vision

some rigidity results for non commutative bernoulli shifts abstract we introduce the outer conjugacy invariants s s s for cocycle actions of discrete groups g on type ii 1 factors n as the set of real numbers t 0 for which the amplification t of can be perturbed to an action respectively to a weakly mixing action we calculate explicitly s s s and the fundamental group of f in the case g has infinite normal subgroups with the relative property t e g when g itself has the property t of kazhdan and is an action of g on the hyperfinite ii 1 factor by connes stormer bernoulli shifts of weights t i i thus s s and f coincide with the multiplicative subgroup s of r generated by the ratios t i t j i j while s z if s 1 i e when all weights are equal and s r otherwise in fact we calculate all the 1 cohomology picture of t t 0 and classify the actions g in terms of their weights t i i in particular we show that any 1 cocycle for g vanishes modulo scalars and that two such actions are cocycle conjugate iff they are conjugate also any cocycle action obtained by reducing a bernoulli action of a group g as above on n g g m n n c tr g to the algebra pnp for p a projection in n p 0 1 cannot be perturbed to a genuine action

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

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16

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

sharpness for c 1 linearization of planar hyperbolic diffeomorphisms abstract c 1 linearization preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions planar hyperbolic diffeomorphisms are the most elementary ones of representatively technical difficulties in the study of c 1 linearization in the poincare domain both eigenvalues inside the unit circle s 1 a lower bound 0 was given such that c 1 smoothness with 0 1 admits c 1 linearization our first purpose of this paper is to prove the sharpness of 0 and give a weaker linearization for 0 in the siegel domain one eigenvalue inside s 1 but the other outside s 1 it is known that c 1 smoothness admits c 1 linearization for all 0 1 the second purpose is to prove that the c 1 linearization is actually a c 1 linearization and give sharp estimates for