new characterizations of morrey spaces and their preduals with applications to fractional laplace equations abstract in this article the authors characterize the morrey spaces as well as their preduals via quadratic functions related to the taylor remainder of the kernel of the riesz potential as applications the authors obtain some strong capacitary inequalities which are then used to study the regularity of the duality weak solution to the fractional laplace equation with measure data

stallings foldings and subgroups of free groups we re cast in a more combinatorial and computational form the topological approach of j stallings to the study of subgroups of free groups

su 2 2 invariant g_2 instantons we initiate the systematic study of g 2 instantons with su 2 2 symmetry as well as developing foundational theory we give existence non existence and classification results for these instantons we particularly focus on mathbb r 4 times s 3 with its two explicitly known distinct holonomy g 2 metrics which have different volume growths at infinity exhibiting the different behaviour of instantons in these settings we also give an explicit example of sequences of g 2 instantons where bubbling and removable singularity phenomena occur in the limit

thompson s group f is not almost convex abstract we show that thompson s group f does not satisfy cannon s almost convexity condition ac n for any positive integer n with respect to the standard generating set with two elements to accomplish this we construct a family of pairs of elements at distance n from the identity and distance 2 from each other which are not connected by a path lying inside the n ball of length less than k for increasingly large k our techniques rely upon fordham s method for calculating the length of a word in f and upon an analysis of the generators geometric actions on the tree pair diagrams representing elements of f

an inner amenable group whose von neumann algebra does not have property gamma we construct inner amenable groups g with infinite conjugacy classes and such that the associated ii1 factor has no non trivial asymptotically central elements i e does not have property gamma of murray and von neumann this solves a problem posed by effros in 1975

on the σ2 invariants of the generalised r thompson groups of type f abstract the 2 invariants of the generalised thompson group f n x 0 x 1 x n x i x j x i n 1 for i j 0 are calculated for n 3 the case n 2 was solved in bieri et al 2010 5

sharp regularity for certain nilpotent group actions on the interval according to the classical plante thurston theorem all nilpotent groups of c 2 diffeomorphisms of the closed interval are abelian using techniques coming from the works of denjoy and pixton farb and franks constructed a faithful action by c 1 diffeomorphisms of 0 1 for every finitely generated torsion free non abelian nilpotent group in this work we give a version of this construction that is sharp in what concerns the holder regularity of the derivatives half of the proof relies on results on random paths on heisenberg like groups that are interesting by themselves

isoperimetric inequalities and the homology of groups an isoperimetric function for a finitely presented group g bounds the number of defining relations needed to prove that a word w g 1 in terms of the length of w suppose g f r is a finitely presented group where f is a finitely generated free group and r is the normal closure of a finite set of relators for each w r we define the area w to be the least number of uses of the given relators needed to prove that w g 1 the isoperimetric function g n is defined to be the maximum of the areas of all the words in r of length n one compares the size of such functions using a suitable order 1 on functions different presentations give rise to equivalent isoperimetric functions where is the equivalence relation determined by 1 interest in such functions was stimulated by gromov 13 who characterized word hyperbolic groups as those with linear isoperimetric function that is g n n further information was obtained by cannon et al 8 who showed that automatic groups satisfy g 1 n and that asynchronously automatic groups satisfy g 1 2 they also showed the free nilpotent group of class 2 and rank 2 has g n n gersten 10 used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups he exploited the fact that for a torsion free one relator groupg the usual 2 dimensional complex having fundamental group isomorphic to g is aspherical the general technique for giving upper bounds for g n is to describe a solution to the word problem for g and analyze its use of defining relations it seems quite difficult to establish lower bounds for g n in this manner in this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for g instead of trying to analyze r directly we consider some of its quotient groups for which additional machinery is available for instance we consider for each w r its image w r f in r r f and define a suitable centralized isoperimetric function g n such that g n 1 g n using the hopf formula it can be shown that r r f is a direct sum of the second homology group h2 g z with a free abelian

a singularity removal theorem for yang mills fields in higher dimensions the purpose of this paper is to investigate the small energy behavior of weakly yang mills fields in rn for n 4 and in particular to extend the singularity removal theorem of uhlenbeck 7 to higher dimensions fix n 4 and let ft be some bounded domain in rn typically we shall restrict our attention to the cubes ft 1 2 n or ft 0 l n let g be a fixed finite dimensional compact lie group it will be convenient to consider g as embedded in some large unitary group u n let g be the lie algebra of g we define a connection on ft to be a section a of t ft q i e a g valued 1 form which is locally l2 for any connection a let 1 f a da aaa

autostackability of thompson s group f abstract the word problem for thompson s group f has a solution but it remains unknown whether f is automatic or has a finite or regular convergent terminating and confluent rewriting system we show that the group f admits a natural extension of these two properties namely autostackability and we give an explicit bounded regular convergent prefix rewriting system for f

diffeomorphism groups of critical regularity let m be a circle or a compact interval and let alpha k tau ge 1 be a real number such that k lfloor alpha rfloor we write mathrm diff alpha m for the group of orientation preserving c k diffeomorphisms of m whose kth derivatives are holder continuous with exponent tau we prove that there exists a continuum of isomorphism types of finitely generated subgroups g le mathrm diff alpha m with the property that g admits no injective homomorphisms into bigcup beta alpha mathrm diff beta m we also show the dual result there exists a continuum of isomorphism types of finitely generated subgroups g of bigcap beta alpha mathrm diff beta m with the property that g admits no injective homomorphisms into mathrm diff alpha m the groups g are constructed so that their commutator groups are simple we give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms for example we show that if alpha ge 1 is a real number not equal to 2 then there is no nontrivial homomorphism mathrm diff alpha s 1 rightarrow bigcup beta alpha mathrm diff beta s 1 finally we obtain an independent result that the class of finitely generated subgroups of mathrm diff 1 m is not closed under taking finite free products

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

new proofs of some results of nielsen given a compact oriented surface m with negative euler characteristic and a homeomorphism z m2 m2 what is the best representative cp m2 m2 in the isotopy class of s thurston s answered this question by defining pseudo anosov homeomorphisms a generalization of anosov diffeomorphisms on t2 which may be characterized as follows the homeomorphism rp is pseudo anosov iff there is a pair of transverse measured geodesic laminations i and au which intersect each closed nonperipheral geodesic in h4 which are topologically preserved by cp and which have their respective transverse measures multiplied under the action of cp by l n and d for some l 1 thurston showed that cp m m2 can be pieced together from homeomorphisms of subsurfaces which are either periodic or pseudoanosov more precisely thurston proved theorem 0 1 the preprint s provides an introduction to this theorem describing many of the useful properties of pseudo anosov diffeomorphisms the proof outlined in s is contained in 111

combinatorial and metric properties of thompson s group t we discuss metric and combinatorial properties of thompson s group t such as the normal forms for elements and uniqueness of tree pair diagrams we relate these properties to those of thompson s group f when possible and highlight combinatorial differences between the two groups we define a set of unique normal forms for elements of t arising from minimal factorizations of elements into convenient pieces we show that the number of carets in a reduced representative of t estimates the word length that f is undistorted in t and that cyclic subgroups of t are undistorted we show that every element of t has a power which is conjugate to an element of f and describe how to recognize torsion elements in t

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

groups of piecewise linear homeomorphisms in this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line we use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them we also identify large simple subgroups in all of them providing examples of finitely presented infinite simple groups

free products in r thompson s group v we investigate some product structures in r thompson s group v primarily by studying the topological dynamics associated with v s action on the cantor set c we draw attention to the class d v c of groups which have embeddings as demonstrative subgroups of v which class can be used to assist in forming various products note that d v c contains all finite groups the free group on two generators and q z and is closed under passing to subgroups and under taking direct products of any member by any finite member if g v and h d v c then g oh embeds into v finally if g h d v c then g h embeds in v using a dynamical approach we also show the perhaps surprising result that z z does not embed in v even though v has many embedded copies of z and has many embedded copies of free products of various pairs of its subgroups

a trace law for the hardy morrey sobolev space abstract this article shows that if h l p and h l q are p hardy morrey space based on the standard lebesgue measure and q hardy morrey space induced by the non negative radon measure respectively then one has the following trace law for the hardy morrey sobolev space i h l p of riesz potentials of order of h l p functions sup 0 f h l p i f h l q f h l p 1 sup x r r n 0 r b x r under 0 n 0 n 0 p n p n 0 q p n p

the dehn function of the generalized thompson group is quadratic abstract guba 8 proved that the dehn function of the thompson group f is quadratic in this paper we apply his idea and extend his result to the generalized thompson group f l for any integer l 3

on the dimension growth of groups we prove that the thompson group f has exponential dimension growth we also prove that every solvable finitely generated subgroup of f has polynomial dimension growth while some elementary amenable subgroups of f and some solvable groups of class 3 have dimension growth at least exp n

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

profinite groups profinite completions and a conjecture of moore abstract let r be any ring with 1 a group and r the corresponding group ring let h be a subgroup of of finite index let m be an r module whose restriction to rh is projective moore s conjecture j pure appl algebra 7 1976 287 assume for every nontrivial element x in at least one of the following two conditions holds m 1 x h e in particular this holds if is torsion free m 2 ord x is finite and invertible in r then m is projective as an r module more generally the conjecture has been formulated for crossed products r and even for strongly graded rings r we prove the conjecture for new families of groups in particular for groups whose profinite completion is torsion free the conjecture can be formulated for profinite modules m over complete groups rings r where r is a profinite ring and a profinite group we prove the conjecture for arbitrary profinite groups this implies serre s theorem on cohomological dimension of profinite groups

complexity and varieties for infinite groups i this two part paper generalizes the usual notion of complexity and varieties for modules over the group algebra of a finite group to a large class of infinite groups the context is modules of typefp for groups in kropholler s classlhf one of the main results is that the category of such modules is generated in a suitable sense by modules induced from finite elementary abelian subgroups this implies that an element of complete cohomology of such a module is nilpotent if and only if its restriction to every finite elementary abelian subgroup is nilpotent it also implies that the complexity of a module of typefp is finite and that the variety is supported on some finite collection of finite elementary abelian subgroups an example is given which shows that the complexity does not determine the rate of growth of the number of generators in a projective resolution in the way it does for finite groups

cone types and geodesic languages for lamplighter groups and thompson s group f we study languages of geodesics in lamplighter groups and thompson s group f we show that the lamplighter groups ln have infinitely many cone types have no regular geodesic languages and have 1 counter context free and counter geodesic languages with respect to certain generating sets we show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context free while with respect to another generating set the full language of geodesics is counter and context free in thompson s group f with respect to the standard finite generating set we show there are infinitely many cone types and that there is no regular language of geodesics we show that the existence of families of seesaw elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set

on weak commutativity in groups abstract for a group g we study homological and homotopical properties of the group g g g g g 1 for g g in particular we show that the operator preserves the soluble of type f p property

the homology of the higman thompson groups we prove that thompson s group mathrm v is acyclic answering a 1992 question of brown in the positive more generally we identify the homology of the higman thompson groups mathrm v n r with the homology of the zeroth component of the infinite loop space of the mod n 1 moore spectrum as mathrm v mathrm v 2 1 we can deduce that this group is acyclic our proof involves establishing homological stability with respect to r as well as a computation of the algebraic k theory of the category of finitely generated free cantor algebras of any type n

tame combing and almost convexity conditions we give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation but which are not minimally almost convex on a standard generating set namely we explicitly construct such combings for thompson s group f and the baumslag solitar groups bs 1 p with p 3 in order to make this construction for thompson s group f we significantly expand the understanding of the cayley complex of this group with respect to the standard finite presentation in particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2 cells adjacent to edges that do not lie on normal form paths

characterizing a free group in its automorphism group for bea and geg if the center of g is trivial then i g a g is one to one and g can be identified with a normal subgroup of a g with the action on g given by conjugation a group g is said to be complete if i g a g is an isomorphism a theorem of burnside ru p 951 says that if g is a group with trivial center then a g is complete if and only if g is a characteristic subgroup of a g let f be a free group of finite rank n 2 in d f joan dyer and 1 used burnside s criterion to show that a f is a complete group more precisely wc showed that if q4 a f a f is an automorphism of a f then d p f

l 2 betti numbers of one relator groups we determine the l 2 betti numbers of all one relator groups and all surface plus one relation groups we also obtain some information about the l 2 cohomology of left orderable groups and deduce the non l 2 result that in any left orderable group of homological dimension one all two generator subgroups are free

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

combinatorial properties of thompson s group f we study some combinatorial consequences of blake fordham s theorems on the word metric of thompson s group f in the standard two generator presentation we explore connections between the tree pair diagram representing an element w of f its normal form in the infinite presentation its word length and minimal length representatives of it we estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates in addition we explore some properties of the cayley graph of f with respect to the two generator finite presentation namely we exhibit the form of dead end elements in this cayley graph and show that it has no deep pockets finally we discuss a simple method for constructing minimal length representatives for strictly positive or negative words

energy identity for stationary yang mills given a principal bundle p rightarrow m over a riemannian manifold with compact structure group g let us consider a stationary yang mills connection a with energy int m f a 2 le lambda if we consider a sequence of such connections a i then it is understood by tian ann math 151 1 193 268 2000 that up to subsequence we can converge a i rightarrow a to a singular limit connection such that the energy measures converge f a i 2 dv g rightarrow f a 2dv g nu where nu e x d lambda n 4 is the n 4 rectifiable defect measure our main result is to show without additional assumptions that for n 4 a e point the energy density e x may be computed explicitly as the sum of the bubble energies arising from blow ups at x each of these bubbles may be realized as a yang mills connection over s 4 itself this energy quantization was proved in riviere commun anal geom 10 4 683 708 2002 assuming a uniform l 1 hessian bound on the curvatures in the sequence in fact our second main theorem is to show this hessian bound holds automatically precisely given a connection a as above we have the a priori estimate int m nabla 2 f a c lambda dim g m for the curvature it is important to note this result is proved in tandem with the energy quantization and not before it indeed we will in fact prove an effective version of the energy identity and it is this effective version which will lead to both the l 1 hessian bound and the classical energy quantization results in the course of the proof we will provide a quantitative version of the bubble tree decomposition which hold in all dimensions with effective estimates for a fixed stationary connections to produce strongest estimates in the paper we introduce an epsilon gauge condition which generalizes the usual coulomb gauge and which will exist with effective control even over singular regions on these epsilon gauges we will provide a new superconvexity estimate which will be a key tool in analyzing higher dimensional annular regions

nonexponential leaves at finite level abstracr previous examples of leaves with nonexponential and nonpolynomial growth due to g hector have occurred at infinite level here the same growth types are produced at finite level in open saturated sets of leaves without holonomy such sets consist of leaves with only one or two locally dense ends and it is shown that the exotic growth types only occur in the case of one locally dense end finally c foliations are produced with open saturated sets as above in which the leaves have strictly fractional growth

approximating nonabelian free groups by groups of homeomorphisms of the real line abstract we show that for a large class c of finitely generated groups of orientation preserving homeomorphisms of the real line the following holds given a group g of rank k in c there is a sequence of k markings g s n n n whose limit in the space of marked groups is the free group of rank k with the standard marking the class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain self similarity type hypothesis examples include thompson s group f higman thompson groups stein thompson groups various bieri strebel groups the golden ratio thompson group and finitely presented nonamenable groups of piecewise projective homeomorphisms for the case of thompson s group f we provide a new and considerably simpler proof of this fact proved by brin in 4

the dehn function of richard thompson s group f is quadratic we prove that the dehn function that is the smallest isoperimetric function of the r thompson s group f is quadratic

stationary biharmonic maps from rm into a riemannian manifold for m 5 we prove that a stationary extrinsic or intrinsic respectively biharmonic map u w2 2 n from rm into a riemnanian manifold n is smooth away from a closed set of m 4 dimensional hausdorff measure zero 2003 wiley periodicals inc

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

metrics and embeddings of generalizations of thompson s group f the distance from the origin in the word metric for generalizations f p of thompson s group f is quasi isometric to the number of carets in the reduced rooted tree diagrams representing the elements of f p this interpretation of the metric is used to prove that every f p admits a quasiisometric embedding into every f q and also to study the behavior of the shift maps under these embeddings the first example of a finitely presented infinite simple group was discovered by r j thompson in 1965 this group g arose in the study of logic and it has many interesting descriptions in 1974 higman 10 used a description as the group of automorphisms of the jonsson tarski algebra to generalize g to an infinite family of groups g has another description as a group of homeomorphisms of the cantor set it has subgroups which may be described as homeomorphism groups of different objects for instance the circle and the unit interval among the people studying these generalizations are brown 3 and bieri and strebel in a set of unpublished notes 1 in this paper we will concentrate on the groups f p which correspond to the groups fp in 3 these families of groups have also been extensively studied by brin and guzman 2 the group f 2 commonly known as thompson s group f can be regarded as the group of piecewise linear orientation preserving homeomorphisms of the interval 0 1 which have breakpoints only at dyadic points and whose slopes where defined are powers of two in 1984 brown and geoghegan 4 found it to be the first example of a finitely presented infinite dimensional torsion free fp group this fact has been extended to all f p in 3 and studied by stein 11 where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered cleary 7 has studied these properties for groups of piecewise linear homeomorphisms with irrational breakpoints and slopes in 5 burillo established the relationship between the word metric of thompson s group f and an estimate of the distance derived from the unique normal form of the elements this algebraic estimate is quasi isometric to the word metric and was used to prove that some subgroups are nondistorted in f in this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams see 6 show that this is quasi isometric to the word metric and use this interpretation to prove that some embeddings of f p into f q including some of those outlined in 2 are quasi isometric the interpretations of these embeddings in terms of received by the editors september 25 1998 and in revised form august 11 1999 2000 mathematics subject classification primary 20f65 secondary 20f05 20f38 20e99 05c25 c 2000 american mathematical society

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

aperiodic pseudorandom number generators based on infinite words in this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators prngs with good statistical behavior we introduce the well distributed occurrences welldoc combinatorial property for infinite words which guarantees absence of the lattice structure defect in related pseudorandom number generators an infinite word u on a d ary alphabet has the welldoc property if for each factor w of u positive integer m and vector v z m d there is an occurrence of w such that the parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m the parikh vector of a finite word v over an alphabet a has its i th component equal to the number of occurrences of the i th letter of a in v we prove that sturmian words and more generally arnoux rauzy words and some morphic images of them have the welldoc property using the testu01 11 and practrand 5 statistical tests we moreover show that not only the lattice structure is absent but also other important properties of prngs are improved when linear congruential generators are combined using infinite words having the welldoc property

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

extensions of amenable groups by recurrent groupoids we show that the amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital schreier graphs this applies to a wide class of groups the amenability of which was an open problem as well as unifies many known examples to one general proof in particular this includes grigorchuk s group basilica group group associated to fibonacci tiling the topological full groups of cantor minimal systems groups acting on rooted trees by bounded automorphisms groups generated by finite automata of linear activity growth and groups naturally appearing in holomorphic dynamics

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

the group of parenthesized braids we investigate a group b that includes artin s braid group b8 and thompson s group f the elements of b are represented by braids diagrams in which the distances between the strands are not uniform and besides the usual crossing generators new rescaling operators shrink or stretch the distances between the strands we prove that b is a group of fractions that it is orderable admits a nontrivial self distributive structure i e one involving the law x yz xy xz it embeds in the mapping class group of a sphere with a cantor set of punctures and that artin s representation of b8 into the automorphisms of a free group extends to b

analytic cycles bott chern forms and singular sets for the yang mills flow on kahler manifolds abstract it is shown that the singular set for the yang mills flow on unstable holomorphic vector bundles over compact kahler manifolds is completely determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle we assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves we derive a singular bott chern formula relating the second chern form of a smooth metric on the bundle to the chern current of an admissible metric on the associated graded sheaf this is used to show that the multiplicities of the top dimensional bubbling locus defined via the yang mills density agree with the corresponding multiplicities for the harder narasimhan seshadri filtration the set theoretic equality of singular sets is a consequence

simple groups separated by finiteness properties we show that for every positive integer n there exists a simple group that is of type mathrm f n 1 but not of type mathrm f n for n ge 3 these groups are the first known examples of this kind they also provide infinitely many quasi isometry classes of finitely presented simple groups the only previously known infinite family of such classes due to caprace remy consists of non affine kac moody groups over finite fields our examples arise from rover nekrashevych groups and contain free abelian groups of infinite rank

homological and finiteness properties of picture groups picture groups are a class of groups introduced by guba and sapir known examples include thompson s groups f t and v in this paper a large class of picture groups is proved to be of type f a morse theoretic argument shows that for a given picture group the rational homology vanishes in almost all dimensions

cohomological finiteness conditions for elementary amenable groups it is proved that every elementary amenable group of type fp admits a cocompact classifying space for proper actions

an algebraic classification of some solvable groups of homeomorphisms abstract we produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group pl o i of piecewise linear orientation preserving homeomorphisms of the unit interval under the operation of composition and also of the generalized r thompson groups f n the first description is as a set of isomorphism classes of groups which is closed under three algebraic operations and the second is as the set of isomorphism classes of subgroups of a countable collection of easily described groups we show the two descriptions are equivalent

cantor systems piecewise translations and simple amenable groups we provide the rst examples of nitely generated simple groups that are amenable and innite to this end we prove that topological full groups of minimal systems are amenable this follows from a general existence result on invariant states for piecewise translations of the integers the states are obtained by constructing a suitable family of densities on the classical bernoulli space

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

the resolution of the yang mills plateau problem in super critical dimensions abstract we study the minimization problem for the yang mills energy under fixed boundary connection in supercritical dimension n 5 we define the natural function space a g in which to formulate this problem in analogy to the space of integral currents used for the classical plateau problem the space a g can be also interpreted as a space of weak connections on a real measure theoretic version of reflexive sheaves from complex geometry we prove the existence of weak solutions to the yang mills plateau problem in the space a g we then prove the optimal regularity result for solutions of this plateau problem on the way to prove this result we establish a coulomb gauge extraction theorem for weak curvatures with small yang mills density this generalizes to the general framework of weak l 2 curvatures previous works of meyer riviere and tao tian in which respectively a strong approximability property and an admissibility property were assumed in addition

dynamical forcing of circular groups in this paper we introduce and study the notion of dynamical forcing basically we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties as an application we show that the set x r z consisting of rotation numbers which can be forced by finitely presented groups is an infinitely generated q module containing countably infinitely many algebraically independent transcendental numbers here a rotation number is forced by a pair g where g is a finitely presented group g and g is some element if the set of rotation numbers of as hom g homeo s 1 is precisely the set 0 we show that the set of subsets of r z which are of the form rot x g r r z r rot hom g homeo s 1 where g varies over countable groups are exactly the set of closed subsets which contain 0 and are invariant under x x moreover we show that every such subset can be approximated from above by rot x g i i for finitely presented g i as another application we construct a finitely generated group which acts faithfully on the circle but which does not admit any faithful c 1 action thus answering in the negative a question of john franks

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

rigidity for c 1 actions on the interval arising from hyperbolicity i solvable groups we consider abelian by cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the abelian subgroup we show that if such a group acts faithfully by c 1 diffeomorphisms of the closed interval with no global fixed point at the interior then the action is topologically conjugate to that of an affine group moreover in case of non abelian image we show a rigidity result concerning the multipliers of the homotheties despite the fact that the conjugacy is not necessarily smooth some consequences for non solvable groups are proposed in particular we give new proofs examples yielding the existence of finitely generated locally indicable groups with no faithful action by c 1 diffeomorphisms of the interval

quasi isometrically embedded subgroups of thompson s groupf abstract the goal of this paper is to study subgroups of thompson s group f which are isomorphic to f z n and f f a result estimating the norm of an element of thompson s group is found and this estimate is used to prove that these particular subgroups are quasi isometrically embedded

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

proving the deterministic period breaking of linear congruential generators using two tile quasicrystals we describe the design of a family of aperiodic prngs aprngs we show how a one dimensional two tile cut and project quasicrystal 2tqc used in conjunction with lcgs in an aprng generates an infinite aperiodic pseudorandom sequence in the suggested design any 2tqc corresponding to unitary quadratic pisot number combined with either one or two different lcgs can be used

some groups of type vf a group is of type vf if it contains a finite index subgroup which has a finite classifying space we construct groups of type vf in which the centralizers of some elements of finite order are not of type vf and groups of type vf containing infinitely many conjugacy classes of finite subgroups from these examples it follows that a group g of type vf need not admit a finite type or finite classifying space for proper actions sometimes also called the universal proper g space we construct groups g for which the minimal dimension of a universal proper g space is strictly greater than the virtual cohomological dimension of g each of our groups embeds in some general linear group over the rational integers applications to algebraic k theory of group algebras and topological k theory of group c algebras are also considered the groups are constructed as finite extensions of bestvina brady groups

automorphisms of generalized thompson groups 0 1 results we study the automorphisms of some generalizations of thompson s groups and their underlying structures the automorphism groups of two of thompson s original groups were analyzed in 2 and were shown to be small and unexotic our results differ sharply from 2 in that we show that the automorphism groups of the generalizations are large and have exotic elements the term exotic is from 1 and is explained later in this introduction richard j thompson introduced the triple of infinite groups f t g in the 1960s and showed that they have several interesting properties these groups were later generalized to infinite groups fn fn tn r gn r for n 2 and r 1 for which f f2 f2 t t2 1 and g g2 1 further background is given in the second part of this introduction the automorphism groups aut f and aut t were analyzed in 2 in this paper we study aut fn aut fn and aut tn n 1 for n 2 we find that all three contain exotic elements and that the complexity increases as n increases we also find that the structures of aut fn and aut tn n 1 are closely related while differing significantly from the structure of aut fn this difference increases as n increases our analysis is only partial in that we discover large subgroups of these groups containing exotic elements but we do not know whether the subgroups 1

topological full groups of one sided shifts of finite type we explore the topological full group g of an essentially principal etale groupoid g on a cantor set when g is minimal we show that g and its certain normal subgroup is a complete invariant for the isomorphism class of the etale groupoid g furthermore when g is either almost finite or purely infinite the commutator subgroup d g is shown to be simple the etale groupoid g arising from a onesided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid for such g g is thought of as a generalization of the higman thompson group we prove that g is of type f and so in particular it is finitely presented this gives us a new infinite family of finitely presented infinite simple groups also the abelianization of g is calculated and described in terms of the homology groups of g