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

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

nonnegative matrix factorization and i divergence alternating minimization abstract in this paper we consider the nonnegative matrix factorization nmf problem given an elementwise nonnegative matrix v r m n find for assigned k nonnegative matrices w r m k and h r k n such that v wh exact nontrivial nonnegative factorizations do not always exist hence it is interesting to pose the approximate nmf problem the criterion which is commonly employed is i divergence between nonnegative matrices the problem becomes that of finding for assigned k the factorization wh closest to v in i divergence an iterative algorithm em like for the construction of the best pair w h has been proposed in the literature in this paper we interpret the algorithm as an alternating minimization procedure a la csiszar tusnady and investigate some of its stability properties nmf is widespreading as a data analysis method in applications for which the positivity constraint is relevant there are other data analysis methods which impose some form of nonnegativity we discuss here the connections between nmf and archetypal analysis

minimal positive realizations of transfer functions with nonnegative multiple poles this note concerns a particular case of the minimality problem in positive system theory a standard result in linear system theory states that any nth order rational transfer function of a discrete time invariant linear single input single output siso system admits a realization of order n in some applications however one is restricted to realizations with nonnegative entries i e a positive system and it is known that this restriction may force the order n of realizations to be strictly larger than n a general solution to the minimality problem i e determining the smallest possible value of n is not known in this note we consider the case of transfer functions with nonnegative multiple poles and give sufficient conditions for the existence of positive realizations of order n n with the help of our results we also give an improvement of an existing result in positive system 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

on the orders of primitive groups with restricted nonabelian composition factors we prove that given c0 there exists c such that the following holds if g is a primitive permutation group of degree n no composition factor of which is an alternating group of degree greater than c0 or a classical group of dimension greater than c0 then g nc in particular if the nonabelian composition factors of g have bounded order then g is polynomially bounded

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

counting arithmetic lattices and surfaces we give estimates on the number al h x of conjugacy classes of arithmetic lattices of covolume at most x in a simple lie group h in particular we obtain a first concrete estimate on the number of arithmetic 3 manifolds of volume at most x our main result is for the classical case h psl 2 where we show that limx log al h x x log x 1 2 the proofs use several different techniques geometric bounding the number of generators of as a function of its covolume number theoretic bounding the number of maximal such and sharp estimates on the character values of the symmetric groups to bound the subgroup growth of

primes in several classes of the positive matrices the classification of primes in the semi ring of the positive matrices is of interest to control and system theory a few examples of primes in the positive matrices are known in this paper results on the classification of primes in the positive matrices in the doubly stochastic matrices and in the doubly stochastic circulants are presented

the cusp amplitudes and quasi level of a congruence subgroup of sl2 over any dedekind domain this is the latest part of an ongoing project aimed at extending algebraic properties of the classical modular group sl2 z to equivalent groups in the theory of drinfeld modules we are especially interested in those properties which are important in the classical theory of modular forms our results are intended to be applicable to the theory of drinfeld modular curves and forms here we are concerned with the cusp amplitudes and level of a subgroup of such a group in particular a congruence subgroup in the process we have discovered that most of the theory of congruence subgroups including the properties of their cusp amplitudes and level can be extended to sl2 d where d is any dedekind ring this means that this theory can be extended to a non arithmetic setting we begin with an ideal theoretic definition of the cusp amplitudes of a subgroup h of sl2 d and extend the remarkable results of larcher for the congruence subgroups of sl2 z we then extend the definition of the cusp amplitude and level of a subgroup h of sl2 d by introducing the notions of quasi amplitude and quasi level quasi amplitudes and quasi level encode more information about h since they are not required to be ideals in general although the level and quasi level can be very different we show that for many congruence subgroups they are equal as a bonus our results provide several new necessary conditions for a subgroup of sl2 d to be a congruence subgroup these include an inequality between the index and level of a congruence subgroup

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

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

k causality coincides with stable causality it is proven that k causality coincides with stable causality and that in a k causal spacetime the relation k coincides with the seifert s relation as a consequence the causal relation the spacetime is strongly causal and the closure of the causal relation is transitive stays between stable causality and causal continuity

global hyperbolicity is stable in the interval topology we prove that global hyperbolicity is stable in the interval topology on the spacetime metrics we also prove that every globally hyperbolic spacetime admits a cauchy hypersurface which remains cauchy under small perturbations of the spacetime metric moreover we prove that if the spacetime admits a complete timelike killing field then the light cones can be widened preserving both global hyperbolicity and the killing property of the field

an efficient algorithm for positive realizations we observe that successive applications of known results from the theory of positive systems lead to an it efficient general algorithm for positive realizations of transfer functions we give two examples to illustrate the algorithm one of which complements an earlier result of cite large finally we improve a lower bound of cite mn2 to indicate that the algorithm is indeed efficient in general

on the group rings of abelian minimax groups ii the singular case abstract the main results of d segal on the group rings of abelian minimax groups j algebra 237 2001 64 94 and of related papers by j e roseblade and c j b brookes are extended to group rings kg where the characteristic of the field k occurs in the spectrum of the abelian minimax group g

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

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

on smooth cauchy hypersurfaces and geroch s splitting theorem given a globally hyperbolic spacetime m we show the existence of a smooth spacelike cauchy hypersurface s and thus a global diffeomorphism between m and r s

methods for scaling to doubly stochastic form abstract new methods for scaling square nonnegative matrices to doubly stochastic form are described a generalized version of the convergence theorem of sinkhorn and knopp 1967 is proved and applied to show convergence for these new methods tests indicate that one of the new methods has significantly better average and worst case behavior than the sinkhorn knopp method for one of the 3 3 examples of marshall and olkin 1968 sk requires 130 times as many operations as the new algorithm to achieve row and column sums 1 10 5

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

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

algebraic structure of classical field theory i kinematics and linearized dynamics for real scalar fields we describe the elements of a novel structural approach to classical field theory inspired by recent developments in perturbative algebraic quantum field theory this approach is local and focuses mainly on the observables over field configurations given by certain spaces of functionals which are studied here in depth the analysis of such functionals is characterized by a combination of geometric analytic and algebraic elements which 1 make our approach closer to quantum field theory 2 allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and 3 provide a new pathway towards understanding dynamics particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations

classification and statistics of finite index subgroups in free products abstract the principal theme of this paper is the enumeration of finite index subgroups in a free product of finite groups under various restrictions on the isomorphism type of in particular we completely resolve the realization asymptotic and distribution problems for free products of cyclic groups of prime order prior to this work these questions were wide open even in the case of the classical modular group this complex of problems usually referred to as poincare klein problem originally arose around 1880 out of the work of klein and poincare on automorphic functions and related number theory but has also grown roots in geometric function theory and more recently in the theory of subgroup growth ideas and techniques from the theory of generalized permutation representations an enumerative theory of wreath product representations recently developed by the first named author play a fundamental role here other tools come from analytic number theory combinatorics and probability theory

counting lattices in simple lie groups the positive characteristic case in this article we prove a conjecture of a lubotzky let g g0 k where k is a local field of characteristic p 5 g0 is a simply connected absolutely almost simple k group of k rank at least 2 we give the rate of growth of x g g a lattice in g vol g x where 1 2 if and only if there is an abstract automorphism of g such that 2 1 we also study the rate of subgroup growth sx of any lattice in g as a result we show that these two functions have the same rate of growth which proves lubotzky s conjecture along the way we also study the rate of growth of the number of equivalence classes of maximal lattices in g with covolume at most x

a polynomial bound for the orders of primitive solvable groups the structure of primitive solvable permutation groups has bee r thoroughly investigated since galois relying on the known structure theorems we determine the largest order a n of a primitive solvable group of degree ii theorem 3 in the course of the proof we obtain the largest order a n of a transitive solvable group of degree n as well theorem 2 although the formula describing the exact value of i is quite complicated we are able to give an asymptotic evaluation it turns out thar the order of a primitive solvable group of degree n is bounded by 24 12 for a constant c 3 24399 theorem i a similar asymptotic estimate obtained recentiy by babai cameron and pilfy 111 shows that the orders of those primitive permutation groups of degree iz in which all nonabelian simple sections have orders less than a iixed integer k are bounded by n where c c k depends on k only related estimates under the assumption of the classification of finite simple groups are given by cameron 2 strong motivation for such asymptotic estimates comes from recent results of luks 151 on the complexity of graph isomorphism testing the results of the present paper have immediate application to graphs of vaience k p we mean that the field multiplication can be defined on the k dimensional vector space over gf p so that this equ ality holds 127 002 8693 82 g g i27 i so2 gg g

level and index in the modular group it is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup this allows a number of existence theorems about non congruence subgroups the level of a subgroup of the modular group can be defined in terms of the action on q we define a similar action to get information on congruence subgroups in fact we get a more powerful result but this appears to be the most useful version

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

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

primes in the semigroup of boolean matrices abstract let a b c be n n matrices of zeros and ones using boolean addition and multiplication we say that a is prime if it is not a permutation matrix and if a bc implies that b or c must be a permutation matrix conditions sufficient for a matrix to be prime are provided and a characterization of primes in terms of a nation of rank is given finally an order property of primes is used to obtain a result on prime factors

arithmetic groups with isomorphic finite quotients abstract two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic consider the map which sends the isomorphism class of an s arithmetic group to the isomorphism class of its profinite completion we show that for a wide class of s arithmetic groups this map is finite to one while the fibers are of unbounded size

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

on the existence of isotropic forms of semi simple algebraic groups over number fields with prescribed local behavior this note is a follow up on the paper a borel g harder existence of discrete cocompact subgroups of reductive groups over local fields j reine angew math 298 1978 53 64 of a borel and g harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi simple group over a number field and considering a suitable s arithmetic subgroup some years ago a lubotzky initiated a program to study the subgroup growth of arithmetic subgroups the current stage of which focuses on counting more precisely determining the asymptotics of the number of lattices of bounded covolume the finiteness of this number was established in a borel g prasad finiteness theorems for discrete subgroups of bounded covolume in semi simple groups publ math inst hautes etudes sci 69 1989 119 171 addendum publ math inst hautes etudes sci 71 1990 173 177 using the formula for the covolume developed in g prasad volumes of s arithmetic quotients of semi simple groups publ math inst hautes etudes sci 69 1989 91 117 work on this program led m belolipetsky and a lubotzky to ask questions about the existence of isotropic forms of semi simple groups over number fields with prescribed local behavior in this paper we will answer these questions a question of similar nature also arose in the work d morris real representations of semisimple lie algebras have forms in proc internat conf on algebraic groups and arithmetic december 17 22 2001 tifr mumbai 2001 pp 469 490 of d morris witte on a completely different topic we will answer that question too

enumerating finite groups of given order let f n denote the number of isomorphism classes of groups of order n let n hw p9ig be the prime factorization of n and let a a n ek and u ii n maxkl1 gi the function f n has been subject to considerable study g higman 15 and c c sims 26 showed that if n is a prime power say n p then f n n 2 27 o l a2 for a x oc p m neumann 22 proved that if the number of isomorphism classes of simple groups of order n is bounded by nca2 for some constant c then f n n 1 c a2 holds in general combining the methods of 22 with some consequences of the classification theorem of finite simple groups a mciver and neumann 21 obtained that in fact the estimates f n na2 2 and f n n1 2 1 2 hold recently d f holt 16 found a rather elegant proof for the bound

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

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

manifolds counting and class field towers in burger et al 2002 12 and goldfeld et al 2004 17 it was conjectured that if h is a simple lie group of real rank at least 2 then the number of conjugacy classes of arithmetic lattices in h of covolume at most x is x h o 1 logx loglogx where h is an explicit constant computable from the absolute root system of h in this paper we prove that this conjecture is false in fact we show that the growth is at rate xclogx a crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of golod and shafarevich on class field towers

on the congruence subgroup problem determination of the metaplectic kernel let f be a global field i e either a number field or a function field in one variable over a finite field and let oo be the set of its archimedean places let g be a finite non empty set of places of f containing oo let o o denote the ring of g integers of f let a denote the ring of g adeles i e the restricted direct product of the completions f v for r eg let f be an absolutely simple simply connected subgroup of sl defined over f recall that a subgroup f of c f is an g arithmetic subgroup if fnsl n o has finite index in f as well as in o fy f c sl n o an g arithmetic subgroup f is a congruence subgroup if there exists a non zero ideal a in o o g such that

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

maximal subgroups in finite and profinite groups we prove that if a finitely generated profinite group g is not generated with positive probability by finitely many random elements then every finite group f is obtained as a quotient of an open subgroup of g the proof involves the study of maximal subgroups of profinite groups as well as techniques from finite permutation groups and finite chevalley groups confirming a conjecture from ann of math 137 1993 203 220 we then prove that a finite group g has at most igic maximal soluble subgroups and show that this result is rather useful in various enumeration problems

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

counting primes groups and manifolds let lambda sl 2 z be the modular group and let c n lambda be the number of congruence subgroups of lambda of index at most n we prove that lim n infinity log c n lambda log n 2 log log n 3 2 sqrt 2 4 the proof is based on the bombieri vinogradov riemann hypothesis on the average and on the solution of a new type of extremal problem in combinatorial number theory similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple lie groups if g is such a lie group and gamma is an irreducible lattice of g it turns out that the subgroup growth of gamma is independent of the lattice and depends only on the lie type of the direct factors of g it can be calculated easily from the root system the most general case of this result relies on the generalized riemann hypothesis but many special cases are unconditional the proofs use techniques from number theory algebraic groups finite group theory and combinatorics

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

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

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

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

counting maximal arithmetic subgroups we study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semisimple lie group using an extension of the method developed by borel and prasad

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