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

trajectory based place recognition for efficient large scale localization place recognition is a core competency for any visual simultaneous localization and mapping system identifying previously visited places enables the creation of globally accurate maps robust relocalization and multi user mapping to match one place to another most state of the art approaches must decide a priori what constitutes a place often in terms of how many consecutive views should overlap or how many consecutive images should be considered together unfortunately such threshold dependencies limit their generality to different types of scenes in this paper we present a placeless place recognition algorithm using a novel match density estimation technique that avoids heuristically discretizing the space instead our approach considers place recognition as a problem of continuous matching between image streams automatically discovering regions of high match density that represent overlapping trajectory segments the algorithm uses well studied statistical tests to identify the relevant matching regions which are subsequently passed to an absolute pose algorithm to recover the geometric alignment we demonstrate the efficiency and accuracy of our methodology on three outdoor sequences including a comprehensive evaluation against ground truth from publicly available datasets that shows our approach outperforms several state of the art algorithms for place recognition furthermore we compare our overall algorithm to the currently best performing system for global localization and show how we outperform the approach on challenging indoor and outdoor datasets

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

a chebyshev criterion with applications abstract we show that a family of certain definite integrals forms a chebyshev system if two families of associated functions appearing in their integrands are chebyshev systems as well we apply this criterion to several examples which appear in the context of perturbations of periodic non autonomous odes to determine bounds on the number of isolated periodic solutions as well as to persistence problems of periodic solutions for perturbed hamiltonian systems

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

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

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

optimized product quantization product quantization pq is an effective vector quantization method a product quantizer can generate an exponentially large codebook at very low memory time cost the essence of pq is to decompose the high dimensional vector space into the cartesian product of subspaces and then quantize these subspaces separately the optimal space decomposition is important for the pq performance but still remains an unaddressed issue in this paper we optimize pq by minimizing quantization distortions w r t the space decomposition and the quantization codebooks we present two novel solutions to this challenging optimization problem the first solution iteratively solves two simpler sub problems the second solution is based on a gaussian assumption and provides theoretical analysis of the optimality we evaluate our optimized product quantizers in three applications i compact encoding for exhaustive ranking 1 ii building inverted multi indexing for non exhaustive search 2 and iii compacting image representations for image retrieval 3 in all applications our optimized product quantizers outperform existing solutions

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

random generation of finite and profinite groups and group enumeration the following article appeared in annals of mathematics 173 2 2011 769 814 and may be found at http annals math princeton edu 2011 173 2 p04

limit cycles for rigid cubic systems we consider planar cubic systems with a unique rest point of center focus type and constant angular velocity for such systems we obtain an affine classification in three families and for two of them their corresponding phase portraits on the poincare sphere we also prove that for two of these families there is uniqueness of limit cycle with respect the third family we give the bifurcation diagram and phase portraits on the poincare sphere of a one parameter sub family exhibiting at least two limit cycles

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

adelic profinite groups let us call a profinite group d adelic if d is a subgroup of the product sl z for some n g 2 one of the main cases of interest is when d p n p is the profinite completion of an s arithmetic subgroup of a semisimple algebraic group let k be an algebraic number field s be a finite set of places of k containing the infinite ones g be a semisimple connected algebraic group over k and g be an s arithmetic group we say that g satisfies the congruence subgroup property if the s congruence kernel s w x 4 c g is a finite group see pr2 chap 9 if n g 3 and s s the s s congruence kernel c sl is trivial and so we can identify the profinite n completion sl z of sl z with the product sl z n n p n p the purpose of this note is twofold firstly we prove some results on bounded generation for an arbitrary finitely generated adelic profinite group then we apply these results to the case of arithmetic groups to prove the following conjecture of a lubotzky conjecture if the profinite completion g of an s arithmetic group g is adelic then g satisfies the congruence subgroup property

an efficient solution to the five point relative pose problem an efficient algorithmic solution to the classical five point relative pose problem is presented the problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points the algorithm consists of computing the coefficients of a tenth degree polynomial in closed form and subsequently finding its roots it is the first algorithm well suited for numerical implementation that also corresponds to the inherent complexity of the problem we investigate the numerical precision of the algorithm we also study its performance under noise in minimal as well as overdetermined cases the performance is compared to that of the well known 8 and 7 point methods and a 6 point scheme the algorithm is used in a robust hypothesize and test framework to estimate structure and motion in real time with low delay the real time system uses solely visual input and has been demonstrated at major conferences

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

commensurability of 1 cusped hyperbolic 3 manifolds we give examples of non fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres in fact we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non monic alexander polynomial

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

invariable generation of infinite groups abstract a subset s of a group g invariably generates g if g s g s s s for each choice of g s g s s in this paper we study invariable generation of infinite groups with emphasis on linear groups our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable we also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements

finite covers of random 3 manifolds a 3 manifold is haken if it contains a topologically essential surface the virtual haken conjecture posits that every irreducible 3 manifold with infinite fundamental group has a finite cover which is haken in this paper we study random 3 manifolds and their finite covers in an attempt to shed light on this difficult question in particular we consider random heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface for this model of random 3 manifold we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds our results contrast with the analogous probabilities for groups coming from random balanced presentations giving quantitative theorems to the effect that 3 manifold groups have many more finite quotients than random groups the next natural question is whether these covers have positive betti number for abelian covers of a fixed type over 3 manifolds of heegaard genus 2 we show that the probability of positive betti number is 0

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

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

product quantization for nearest neighbor search this paper introduces a product quantization based approach for approximate nearest neighbor search the idea is to decompose the space into a cartesian product of low dimensional subspaces and to quantize each subspace separately a vector is represented by a short code composed of its subspace quantization indices the euclidean distance between two vectors can be efficiently estimated from their codes an asymmetric version increases precision as it computes the approximate distance between a vector and a code experimental results show that our approach searches for nearest neighbors efficiently in particular in combination with an inverted file system results for sift and gist image descriptors show excellent search accuracy outperforming three state of the art approaches the scalability of our approach is validated on a data set of two billion vectors

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

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

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

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

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

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