on the degree of groups of polynomial subgroup growth let g be a finitely generated residually finite group and let an g denote the number of index n subgroups of g if an g nr for some a and for all n then g is said to have polynomial subgroup growth psg for short the degree of g is then defined by deg g lim sup 1log an g very little seems to be known about the relation between deg g and the algebraic structure of g we derive a formula for computing the degree of certain metabelian groups which serves as a main tool in this paper addressing a problem posed by lubotzky we also show that if h g is a finite index subgroup then deg g deg h 1 a large part of the paper is devoted to the structure of groups of small degree we show that an g is bounded above by a linear function of n if and only if g is virtually cyclic we then determine all groups of degree less than 3 2 and reveal some connections with plane crystallographic groups it follows from our results that the degree of a finitely generated group cannot lie in the open interval 1 3 2 our methods axe largely number theoretic and density theorems a la chebotarev play essential role in the proofs most of the results also rely implicitly on the classification of finite simple groups

extreme growth rate of periodic orbits for equivalent differentiable flows abstract we prove that the extreme growth rate of periodic orbits of differentiable flows is preserved by lipschitz equivalence when all the fixed points are hyperbolic and thus it is an invariant in an open dense subset x h r of the set of c r 1 r flows by contrast for any 1 r there exist two infinite dimensional connected subsets of equivalent c r flows disjoint from x h r such that the growth rate of periodic orbits attains infinity and zero respectively

strong approximation for zariski dense subgroups of semi simple algebraic groups 1 1 theorem let k be an algebraically closed field of characteristic different from 2 and 3 and g an almost simple connected and simply connected algebraic group defined over k let f be a finitely generated zariski dense subgroup of g k and a the subring of k generated by the traces tr ad y y e p then there exist b e a a subgroup f of finite index in f and a structure gab of a group scheme over ab on g such that f c ga ab and f is dense in gala b a

on the number of conjugacy classes of finite nilpotent groups abstract we establish the first super logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group in particular we obtain that for any constant c there are only finitely many finite p groups of order p m with at most c m conjugacy classes this answers a question of l pyber

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

a new class of multilevel decomposition algorithms for non monotone problems based on the quasidifferentiability concept abstract a convex multilevel decomposition approach is proposed for the solution of static analysis problems involving non monotone possibly multivalued laws the theory is developed here for model problems of structures having non monotone interface or boundary conditions first we decompose appropriately the non monotone laws writing them as a difference of monotone constituents in the general case this is related to the quasidifferentiability concept this permits us to obtain a system of convex variational inequalities the solution s of which describe the position s of static equilibrium of the considered structure then the problems are formulated as min min problems for appropriately defined lagrangian functions convex optimization algorithms of various complexity are used in a multilevel scheme for the numerical solution of the considered structural analysis problem numerical results concerning the calculation of an elastic contact problem and an elastic stamp problem illustrate the theory

approximate subgroups of residually nilpotent groups we show that a k approximate subgroup a of a residually nilpotent group g is contained in boundedly many cosets of a finite by nilpotent subgroup the nilpotent factor of which is of bounded step combined with an earlier result of the author this implies that a is contained in boundedly many translates of a coset nilprogression of bounded rank and step the bounds are effective and depend only on k in particular if g is nilpotent they do not depend on the step of g as an application we show that there is some absolute constant c such that if g is a residually nilpotent group and if there is an integer n 1 such that the ball of radius n in some cayley graph of g has cardinality bounded by n c log log n then g is virtually log n step nilpotent

enveloping algebras satisfying a polynomial identity in this paper we study ordinary and restricted enveloping algebras satisfying a polynomial identity we first show how the methods of j bergen and d s passman j algebra in press can be used to handle u l in all characteristics and u l if the latter ring is prime in particular we offer a simpler proof of the result of yu a bachturin j austral math soc 18 1974 10 21 on ordinary enveloping algebras in characteristic p 0 our main theorem asserts that in general u l is p i if and only if l has a restricted subalgebra of finite codimension which is essentially commutative these results are clearly the lie algebra analogs of the p i group ring theorems of m k smith j algebra 18 1971 477 499 and d s passman pacific j math 36 1971 467 483

character degrees of p groups and pro p groups in the 1970s isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p group g with respect to the number of different irreducible character degrees of g so far there are just a few partial results for this conjecture in this note we say that a pro p group g has property i if there is a real number d d g that just depends on g such that for any open normal subgroup n d l g n log2 cd g n d we prove that any p adic analytic pro p group has property i we also study the first congruence subgroup g of a classical chevalley group g with respect to the local ring fpt we show that if lie g fp has a non degenerated killing form then g has property i

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

pro finite p adic lie algebras abstract let p be a prime number a finite nilpotent lie ring of characteristic a power of p is called finite p a pro p lie ring is an inverse limit of finite p lie rings pro p lie rings play a role in lie theory similar to that played by pro p groups in group theory every pro p lie ring admits the structure of a lie algebra over the p adic integers furthermore every p adic lie algebra of finite rank as a p adic module has an open pro p subalgebra we make a detailed study of pro p lie rings in terms of various properties including their topology prufer rank subring growth and p adic module structure in particular we prove the equivalence of the following conditions for a finitely generated pro p lie ring l l has finite prufer rank l is isomorphic to a closed subring of gl v for some p adic module v of finite rank and for sufficiently large n the lie f p subalgebra w n e 12 t e 22 gl 2 f p t t n is not an open section of l by reducing to the pro p lie ring case we also prove that all engelian pro finite lie rings are locally nilpotent this is a lie theoretic analogue of zelmanov s theorem which states that every periodic pro p group is locally finite

on pro p groups with potent filtrations abstract in this note we prove that if g is pf group of finite exponent then the exponent of the second homology group h 2 g m divides the exponent of g for every profinite trivial z g module m we introduce the notion of the exponential rank of a pro p group and find a bound for the exponential rank of a pf group

topological entropy of fixed point free flows topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy topological entropy for one parameter flows on a compact metric spaces is defined by bowen general statements are proved about this entropy but it is not easy to calculate the topological entropy and to show it is invariant under conjugacy for all this i would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits some well known results are proved as well using this definition these results enable us to prove some results which seem difficult to prove using bowen s definition also we show here that this definition is equivalent to bowen s definition for any flow without fixed points on a compact metric space finally it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections

group presentation p adic analytic groups and lattices in mathrm sl _2 mathbf c groups by using their pro p completions this method is based on the observation that if x r is a presentation of the finitely generated f g group f then x r considered as a presentation for a pro p group is actually a presentation for t the pro p completion of f first we prove a generalization of a theorem of golod and shafarevich cf h theorem let g be a p adic analytic pro p group different from zp the p adic integers if x r is a minimal presentation of g then 1x 2 iri 2

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

topological entropies of equivalent smooth flows we construct two equivalent smooth flows one of which has positive topological entropy and the other has zero topological entropy this provides a negative answer to a problem posed by ohno

nonmatrix varieties and nil generated algebras whose units satisfy a group identity abstract letr denote the group of units of an associative algebrarover an infinite fieldf we prove that ifris unitarily generated by its nilpotent elements thenr satisfies a group identity precisely whenrsatisfies a nonmatrix polynomial identity as an application we examine the group algebrafgof a torsion groupgand the restricted enveloping algebrau l of ap nil restricted lie algebral giambruno sehgal and valenti recently proved that if the group of units fg satisfies a group identity thenfgsatisfies a polynomial identity thus confirming a conjecture of brian hartley we show that in fact fg satisfies a group identity if and only iffgsatisfies a nonmatrix polynomial identity in the case of restricted enveloping algebras we prove thatu l satisfies a group identity if and only ifu l satisfies the engel condition

a new variational principle and duality for periodic solutions of hamilton s equations we are dealing with a classical boundary value problem concerning periodic solutions of hamiltonian equations p f k f 4th p f i f h x t p t 1 where h r x r x r r is a given function called the hamiltonian in mechanics 1 and also in mathematics 171 with a given hamiltonian the lagrangian l defined by the legendre transform l 4 x a sup k p wc x pi d 2

asymptotic growth of finite groups abstract we investigate word growth and structure of certain infinite families of finite groups this work is motivated by results of bass wolf milnor gromov and grigorchuk on the word growth and structure of infinite groups we define word growth for families of finite groups and prove structure theorems relating their growth types to their group structures some results are analogous to the infinite cases however differences are also noted as well as other results

critical point approximation through exact regularization construction d algorithmes iteratifs pour la resolution des problemes extremaux non differentiables regularisation d une fonctionnelle application aux problemes differentiels semi elliptiques en physique des plasmas et en mecanique des fluides

characterization of p adic analytic groups in terms of wreath products abstract we show that a finitely generated pro p group is p adic analytic i e can be given the structure of a lie group over q p if and only if it does not involve arbitrarily large wreath products of the form c p wr c p n this result whose proof applies zelmanov s recent solution to the restricted burnside problem is in fact equivalent to zelmanov s theorem

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

finitely generated groups p adic analytic groups and poincare series this is equivalent to the coefficients apn g satisfing a linear recurrence relation with constant coefficients for sufficiently large n these poincare series were first studied in the case when g is a finitely generated torsion free nilpotent group in a paper by grunewald segal and smith gssm there the authors established that for such a group g p s is a rational function in p8s in that setting the functions g p s are the local factors associated with the dirichlet series

covering spaces of 3 orbifolds let o be a compact orientable 3 orbifold with non empty singular locus and a finite volume hyperbolic structure equivalently the interior of o is the quotient of hyperbolic 3 space by a lattice in psl 2 c with torsion then we prove that o has a tower of finite sheeted covers oi with linear growth of p homology for some prime p this means that the dimension of the first homology with mod p coefficients of the fundamental group of oi grows linearly in the covering degree the proof combines techniques from 3 manifold theory with group theoretic methods including the golodshafarevich inequality and results about p adic analytic pro p groups this has several consequences firstly the fundamental group of o has at least exponential subgroup growth secondly the covers oi have positive heegaard gradient thirdly we use it to show that a group theoretic conjecture of lubotzky and zelmanov would imply that o has large fundamental group this implication uses a new theorem of the author which will appear in a forthcoming paper these results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3 orbifold with non empty singular locus has large fundamental group many of the above results apply also to 3 manifolds commensurable with an orientable finite volume hyperbolic 3 orbifold with non empty singular locus this includes all closed orientable hyperbolic 3 manifolds with rank two fundamental group and all arithmetic 3 manifolds

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

convex multilevel decomposition algorithms for non monotone problems a convex multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non monotone possibly multivalued laws the theory is developed here for a model structure with non monotone interface or boundary conditions first the non monotone laws are written in the form of a difference of two monotone functions under this decomposition the non linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non convex min min problems for appropriately defined lagrangian functions the solution s of each one of the aforementioned problems describe the position s of static equilibrium of the considered structure in this paper a multilevel optimization scheme due to auchmuty 1 is used for the numerical solution of the problem the most interesting feature of this method from the computational mechanics standpoint is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem or in terms of mechanics an appropriately modified monotone unilateral problem thus existing algorithms and software can be used for the numerical solution with minor modifications numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non monotone stick slip frictional effects illustrate the theory

dimension subgroups nilpotency indices and the number of generators of ideals in p group algebras abstract various problems in modular p group algebras are solved through extensive study of dimension subgroups as an application it is shown that if g is an infinite res p group p 2 and k is a field of characteristic p then the least upper bound on the numbers of generators of ideals in kg equals the minimal index of a cyclic subgroup of g

new lower bounds on subgroup growth and homology growth subgroup growth is an important new area of group theory it attempts to quantify the number of finite index subgroups of a group as a function of their index in this paper we will provide new strong lower bounds on the subgroup growth of a variety of different groups this will include the fundamental groups of all finite volume hyperbolic 3 manifolds by using the correspondence between subgroups and covering spaces we will be able to address the following natural question how many finite sheeted covering spaces does a hyperbolic 3 manifold have as a function of the covering degree

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

on livsic s theorem superrigidity and anosov actions of semisimple lie groups we prove a generalization of livsic s theorem on the vanishing of the cohomology of certain types of dynamical systems as a consequence we strengthen a result due to zimmer concerning algebraic hulls of anosov actions of semisimple lie groups combining this with topological superrigidity we find a holder geometric structure for multiplicity free anosov actions

concave duality application to problems dealing with difference of functions in this paper a duality is introduced in a concave sense and its relationship with toland s duality is studied along with several formulas dealing with the conjugates of differences of convex functions

a finiteness condition on subgroups of large derived length abstract the authors discuss the class s d r of groups in which every finitely generated subgroup is either at most r generated or soluble of derived length at most d such groups need not be of finite rank or soluble of derived length at most d in general a structure theorem is obtained for locally finite and for certain locally nilpotent s d r groups

congruence properties of zariski dense subgroups i this paper deals with the following general situation we are given an algebraic group g defined over a number field k and a subgroup f of the group g k of k rational points of g then what should it mean for f to be a large subgroup we might require f to be a lattice in g to be arithmetic to contain many elements of a specific kind to have a large closure in some natural topology et cetera there are many theorems proving implications between conditions of size of this kind we shall consider the case of g a k simple group usually q simple the zariski topology of g for which the closed sets are those defined by the vanishing of polynomial functions is very coarse on the other hand the various valuations v of k each give rise to a strong topology on g k if v is non archimedean and kv is the completion of k with respect to v then g kv is a non archimedean lie group and if is the ring of integers of kv the open subgroup g 5 j of g kv of integral points is defined for all but finitely many v the zariski closure f of a subgroup f of g k is a c algebraic subgroup of g while the strong closure fy is a lie subgroup of g kv for elementary reasons the dimension of tv is at most the dimension of f our results are in the opposite direction we show under suitable conditions on g that if f g then for almost all v we have that tv contains g ov to illustrate this with a specific example if ml 5 mk are matrices in sl2 q generating a group f which is zariskidense in sl2 then for all sufficiently large prime numbers p we have f p sl2 zp further this result is effective in the sense that we could in principle check the hypothesis for given mx mk and derive a bound on p beyond which the conclusion holds it is perhaps worth remarking that this special case which is the simplest application of our results may be proved with much less effort than the general theorem

existence and regularity in the small in the calculus of variations abstract a local existence theorem is proved for the basic problem in the calculus of variations that of minimizing l t x x dt over a class of functions x assuming given boundary conditions the lagrangian l is only assumed to be locally lipschitz and strictly convex in its x variable

on groups of polynomial subgroup growth let be a finitely generated group andan the number of its subgroups of indexn we prove that assuming is residually nilpotent e g linear thenan grows polynomially if and only if is solvable of finite rank this answers a question of segal the proof uses a new characterization ofp adic analytic groups the theory of algebraic groups and the prime number theorem the method can be applied also to groups of polynomial word growth

the bounded complexity function versus the unbounded complexity function abstract we construct two point wise periodic flows which are equivalent such that all the complexity functions of one flow are bounded while the other flow has an unbounded complexity function

quasi shadowing and limit quasi shadowing for quasi partially hyperbolic strings of flows abstract in this paper we study the quasi shadowing property for quasi partially hyperbolic pseudo orbit made by quasi partially hyperbolic strings we also investigate the corresponding l p limit and asymptotic quasi shadowing properties as an application we prove that the set of points with periodic center leaves is dense in the non wondering set for some partially hyperbolic flows

generating pairs for finite index subgroups of sl n z abstract let n 3 lubotzky 2 asked if every finite index subgroup of sl n z contains a finite index subgroup which is generated by two elements venkataramana 11 proved that every finite index subgroup of sl n z contains a finite index subgroup which is generated by three elements since then it was widely believed that the answer to lubotzky s question is positive and we show that this is indeed the case in fact we prove a stronger statement for almost every element g sl n z there exists h sl n z such that sl n z g h m for every m 1

duality for non convex variational principles abstract this paper treats the construction of dual variational principles for non convex problems using lagrangian methods besides the well known saddle point theory there are dualities involving minimizing the lagrangian in both variables and dualities where one first maximizes and then minimizes the lagrangian in each variable when the primal problem has certain structure an associated canonical lagrangian is defined which leads to the appropriate dual variational principle and to a hamiltonian description the properties of and correspondence between the critical points are analyzed the last five sections are devoted to examples of these dual principles

pro p groups with constant generating number on open subgroups abstract let p be a prime we classify finitely generated pro p groups g which satisfy d h d g for all open subgroups h of g here d h denotes the minimal number of topological generators for the subgroup h within the category of p adic analytic pro p groups this answers a question of iwasawa

sieve methods in group theory i powers in linear groups a general sieve method for groups is formulated it enables one to measure subsets of a finitely generated group as an application we show that if is a finitely generated non virtually solvable linear group of characteristic zero then the set of proper powers in is exponentially small this is a far reaching strengthening of the main result of hkls

asymptotic results for primitive permutation groups abstract we prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostnc n where n denotes the maximal exponent occurring in the prime factorization ofn this result is applied to investigating maximal subgroup growth of infinite groups we then proceed by showing that if the point stabilizerg of a primitive groupgof degreendoes not have the alternating group alt d as a section then the order ofgis bounded by a polynomial inn this result extends a well known theorem of babai cameron and palfy it is used to prove for example that ifhis a subgroup of indexnin a groupg andhis a product ofbcyclic groups then g hg ncwherecdepends onb

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

lie solvable enveloping algebras of characteristic two abstract lie solvable restricted enveloping algebras were characterized by riley and shalev except when the ground field is of characteristic 2 we resolve the characteristic 2 case here which completes the classification as an application of our result we obtain a characterization of ordinary lie algebras over any field whose enveloping algebra is lie solvable

variational principles for operator equations and initial value problems on decrit et on etudie des principes variationnels pour certaines classes d equations operateur et de problemes aux valeurs initiales

compact groups in which all elements have countable right engel sinks a right engel sink of an element g of a group g is a set mathscr r g such that for every x in g all sufficiently long commutators g x x dots x belong to mathscr r g thus g is a right engel element precisely when we can choose mathscr r g 1 it is proved that if every element of a compact hausdorff group g has a countable right engel sink then g has a finite normal subgroup n such that g n is locally nilpotent

on uniform exponential growth for linear groups let be a finitely generated group given a finite set of generators s of the word length ls for an element is defined to be the smallest positive integer for which there exist s1 sn s s 1 such that s1 sn for each n n denote by bs n the set of elements in whose word length with respect to s is at most n it follows from the subadditive property of ls that limn bs n 1 n exists which we denote by s a finitely generated group is said to be of exponential growth if s 1 of polynomial growth if for some c 0 and d n bs n c n d for all n 1 and of intermediate growth otherwise for some finite generating set s of observe that the growth type of does not depend on the choice of generating set s if a finitely generated group is linear it is known that is either of polynomial growth in which case is virtually nilpotent or of exponential growth otherwise tit72 mil68 wol68