a note on set theoretic solutions of the yang baxter equation abstract this paper shows that every finite non degenerate involutive set theoretic solution x r of the yang baxter equation whose permutation group g x r has cardinality which is a cube free number is a multipermutation solution some properties of finite braces are also investigated it is also shown that if a is a left brace whose cardinality is an odd number and a b a b for all a b a then a is a two sided brace and hence a jacobson radical ring it is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level

metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

matched pairs approach to set theoretic solutions of the yang baxter equation we study set theoretic solutions x r of the yang baxter equations on a set x in terms of the induced left and right actions of x on itself we give a characterisation of involutive square free solutions in terms of cyclicity conditions we characterise general solutions in terms of abstract matched pair properties of the associated monoid s x r and we show that r extends as a solution rs on s x r as a set finally we study extensions of solutions both directly and in terms of matched pairs of their associated monoids we also prove several general results about matched pairs of monoids s of the required type including iterated products s s s equivalent to rs a solution and extensions s t rs t examples include a general double construction s s rs s and some concrete extensions their actions and graphs based on small sets

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

a family of solutions of the yang baxter equation a new method to construct involutive non degenerate set theoretic solutions xn r n of the yang baxter equation for any positive integer n from a given solution x r is presented furthermore the permutation group g xn r n associated with the solution xn r n is isomorphic to a subgroup of g x r and in many cases g xn r n g x r

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

multipermutation solutions of the yang baxter equation set theoretic solutions of the yang baxter equation form a meeting ground of mathematical physics algebra and combinatorics such a solution consists of a set x and a function r x x x x which satisfies the braid relation

factorizations of skew braces we introduce strong left ideals of skew braces and prove that they produce non trivial decomposition of set theoretic solutions of the yang baxter equation we study factorization of skew left braces through strong left ideals and we prove analogs of ito s theorem in the context of skew left braces as a corollary we obtain applications to the retractability problem of involutive non degenerate solutions of the yang baxter equation finally we classify skew braces that contain no non trivial proper characteristic ideals

lower bounds of the laplacian spectrum of graphs based on diameter let g be a connected graph of order n the diameter of g is the maximum distance between any two vertices of g in the paper we will give some lower bounds for the algebraic connectivity and the laplacian spectral radius of g in terms of the diameter of g

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

quadratic algebras of skew type and the underlying monoids abstract we consider algebras over a field k defined by a presentation k x1 xn r where r consists of n 2 square free relations of the form xixj xkxl with every monomial xixj i j appearing in one of the relations certain sufficient conditions for the algebra to be noetherian and pi are determined for this we prove more generally that right noetherian algebras of finite gelfand kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity the structure of the underlying monoid defined by the same presentation is described this is used to derive information on the prime radical and minimal prime ideals some examples are described in detail earlier gateva ivanova and van den bergh and jespers and okninski considered special classes of such algebras in the contexts of noetherian algebras grobner bases finitely generated solvable groups semigroup algebras and set theoretic solutions of the yang baxter equation

retractability of set theoretic solutions of the yang baxter equation abstract it is shown that square free set theoretic involutive non degenerate solutions of the yang baxter equation whose associated permutation group referred to as an involutive yang baxter group is abelian are retractable in the sense of etingof schedler and soloviev this solves a problem of gateva ivanova in the case of abelian iyb groups it also implies that the corresponding finitely presented abelian by finite groups called the structure groups are poly z groups secondly an example of a solution with an abelian involutive yang baxter group which is not a generalized twisted union is constructed this answers in the negative another problem of gateva ivanova the constructed solution is of multipermutation level 3 retractability of solutions is also proved in the case where the natural generators of the iyb group are cyclic permutations moreover it is shown that such solutions are generalized twisted unions

symmetric patterns of geodesics and automorphisms of surface groups we prove a non equivariant version of mostow rigidity for symmetric patterns of geodesics in hyperbolic space this result allows for a classification of pseudo anosov surface group automorphisms based on the large scale geometric structure of their orbits

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

solutions of the yang baxter equation associated with a left brace abstract given a left brace b a method is given to construct explicitly all the non degenerate involutive set theoretic solutions x r of the yang baxter equation such that the associated permutation group g x r is isomorphic as a left brace to b this method depends entirely on the brace structure of b

braces and the yang baxter equation several aspects of relations between braces and non degenerate involutive set theoretic solutions of the yang baxter equation are discussed and many consequences are derived in particular for each positive integer n a finite square free multipermutation solution of the yang baxter equation with multipermutation level n and an abelian involutive yang baxter group is constructed this answers a problem of gateva ivanova and cameron it is proved that finite non degenerate involutive set theoretic solutions of the yang baxter equation whose associated involutive yang baxter group is abelian are multipermutation solutions earlier the authors proved this with the additional square free hypothesis on the solutions it is also proved that finite square free non degenerate involutive set theoretic solutions associated to a left brace are multipermutation solutions

combinatorial solutions to the reflection equation abstract we use ring theoretic methods and methods from the theory of skew braces to produce set theoretic solutions to the reflection equation we also use set theoretic solutions to construct solutions of the parameter dependent reflection equation

group splittings and asymptotic topology it is a consequence of the theorem of stallings on groups with many ends that splittings over finite groups are preserved by quasi isometries in this paper we use asymptotic topology to show that group splittings are preserved by quasi isometries in many cases roughly speaking we show that splittings are preserved under quasi isometries when the vertex groups are fundamental groups of aspherical manifolds or more generally coarse pd n groups and the edge groups are smaller than the vertex groups

skew polynomial rings with binomial relations abstract in this paper we continue the study of a class of standard finitely presented quadratic algebrasaover a fixed fieldk called binomial skew polynomial rings we consider some combinatorial properties of the set of defining relationsfand their implications for the algebraic properties ofa we impose a condition called onfand prove that in this caseais a free module of finite rank over a strictly ordered noetherian domain we show that an analogue of the diamond lemma is true for one sided ideals of a skew polynomial ringawith condition we prove also that if the set of defining relationsfis square free then condition is necessary and sufficient for the existence of a finite groebner basis of every one sided ideal ina and for left and right noetherianness ofa as a corollary we find a class of finitely generated non commutative semigroups which are left and right noetherian

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

set theoretic solutions of the yang baxter equation braces and symmetric groups abstract we involve simultaneously the theory of braided groups and the theory of braces to study set theoretic solutions of the yang baxter equation ybe we show the intimate relation between the notions of a symmetric group in the sense of takeuchi i e a braided involutive group and a left brace we find new results on symmetric groups of finite multipermutation level and the corresponding braces we introduce a new invariant of a symmetric group g r the derived chain of ideals of g which gives a precise information about the recursive process of retraction of g we prove that every symmetric group g r of finite multipermutation level m is a solvable group of solvable length m to each set theoretic solution x r of ybe we associate two invariant sequences of involutive braided groups i the sequence of its derived symmetric groups ii the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction of x r we find new criteria necessary and sufficient to claim that x r is a multipermutation solution

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

a lower bound for algebraic connectivity based on the connection graph stability method this paper introduces the connection graph stability method and uses it to establish a new lower bound on the algebraic connectivity of graphs the second smallest eigenvalue of the laplacian matrix of the graph that is sharper than the previously published bounds the connection graph stability score for each edge is defined as the sum of the lengths of the shortest paths making use of that edge we prove that the algebraic connectivity of the graph is bounded below by the size of the graph divided by the maximum connection graph stability score assigned to the edges

homology of left non degenerate set theoretic solutions to the yang baxter equation abstract this paper deals with left non degenerate set theoretic solutions to the yang baxter equation lnd solutions a vast class of algebraic structures encompassing groups racks and cycle sets to each such solution there is associated a shelf i e a self distributive structure which captures its major properties we consider two co homology theories for lnd solutions one of which was previously known in a reduced form for biracks only an explicit isomorphism between these theories is described for groups and racks we recover their classical co homology whereas for cycle sets we get new constructions for a certain type of lnd solutions including quandles and non degenerate cycle sets the co homologies split into the degenerate and the normalized parts we express 2 cocycles of our theories in terms of group cohomology and in the case of cycle sets establish connexions with extensions this leads to a construction of cycle sets with interesting properties

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

connection graph stability method for synchronized coupled chaotic systems this paper elucidates the relation between network dynamics and graph theory a new general method to determine global stability of total synchronization in networks with different topologies is proposed this method combines the lyapunov function approach with graph theoretical reasoning in this context the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph in particular the method is applied to the study of synchronization in rings of 2k nearest neighbor coupled oscillators a rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators this bound is explicitly linked with the average path length of the coupling graph contrary to the master stability function approach developed by pecora and carroll the connection graph stability method leads to global stability of synchronization and it permits not only constant but also time dependent interaction coefficients in a companion paper blinking model and synchronization in small world networks with a time varying coupling see this issue this method is extended to the blinking model of small world networks where in addition to the fixed 2 k nearest neighbor interactions all the remaining links are rapidly switched on and off independently of each other

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

braces radical rings and the quantum yang baxter equation abstract non degenerate cycle sets are equivalent to non degenerate unitary set theoretical solutions of the quantum yang baxter equation we embed such cycle sets into generalized radical rings braces and study their interaction in this context we establish a galois theory between ideals of braces and quotient cycle sets our main result determines the relationship between two square free cycle sets operating transitively on each other