local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

a product theorem in free groups if a is a nite subset of a free group with at least two noncommuting elements thenja a aj jaj 2 logjaj o 1 more generally the same conclusion holds in an arbitrary virtually free group unless a generates a virtually cyclic subgroup the central part of the proof of this result is carried on by estimating the number of collisions in multiple products a1 ak we include a few simple observations showing that in this statistical context

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

verlet i r respa impulse is limited by nonlinear instabilities this paper shows that in molecular dynamics md when constant energy nve simulations of newton s equations of motion are attempted using the multiple time stepping mts integrator verlet i r respa impulse there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period s of the fastest motion s in the system this is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem the numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein but also the constrained dynamics simulation of a solvated protein representing a range of simulation protocols commonly in use by biomolecular modelers the observed and predicted instabilities match exactly previous work has identified and explained a linear instability for verlet i r respa impulse at around half the period of the fastest motion mandziuk and schlick discovered nonlinear resonances in single time stepping md integrators but unstable nonlinear resonances for mts integrators are reported here for the first time this paper also offers an explanation on the instability of mts constrained molecular dynamics simulations of explicitly solvated proteins more aggressive multiple step sizes are possible with mild langevin coupling or targeted langevin coupling and its combination with the mollified impulse method permits step sizes 3 to 4 times larger thanverlet i r respa impulse while still retaining some accuracy

expansion in sl d z q z q arbitrary let s be a fixed finite symmetric subset of sl d z and assume that it generates a zariski dense subgroup g we show that the cayley graphs of q g with respect to the generating set q s form a family of expanders where q is the projection map z z q z

on threefolds isogenous to a product of curves abstract a threefold isogenous to a product of curves x is a quotient of a product of three compact riemann surfaces of genus at least two by the free action of a finite group in this paper we study these threefolds under the assumption that the group acts diagonally on the product we show that the classification of these threefolds is a finite problem present an algorithm to classify them for a fixed value of o x and explain a method to determine their hodge numbers running an implementation of the algorithm we achieve the full classification of threefolds isogenous to a product of curves with o x 1 under the assumption that the group acts faithfully on each factor

nilprogressions and groups with moderate growth abstract we show that doubling at some large scale in a cayley graph implies uniform doubling at all subsequent scales the proof is based on the structure theorem for approximate subgroups proved by green tao and the first author we also give a number of applications to the geometry and spectrum of finite cayley graphs for example we show that a finite group has moderate growth in the sense of diaconis and saloff coste if and only if its diameter is larger than a fixed power of the cardinality of the group we call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter we also give bounds on the cheeger constant first eigenvalue of the laplacian and mixing time this can be seen as a finite group version of gromov s theorem on groups with polynomial growth it also improves on a result of lackenby regarding property in towers of coverings another consequence is a universal upper bound on the diameter of all finite simple groups independent of the cfsg

uniform expansion bounds for cayley graphs of sl2 fp we prove that cayley graphs of sl2 fp are expanders with respect to the projection of any fixed elements in sl 2 z generating a non elementary subgroup and with respect to generators chosen at random in sl2 fp

on the local global conjecture for integral apollonian gaskets we prove that a set of density one satisfies the local global conjecture for integral apollonian gaskets that is for a fixed integral primitive apollonian gasket almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

the affine sieve we establish the main saturation conjecture connected with executing a brun sieve in the setting of an orbit of a group of affine linear transformations this is carried out under the condition that the zariski closure of the group is levi semisimple it is likely that this condition is also necessary for such saturation to hold 2013 american mathematical society

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

on the diameter of permutation groups given a finite group g and a set a of generators the diameter diam g g a of the cayley graph g g a is the smallest l such that every element of g can be expressed as a word of length at most l in a a 1 we are concerned with bounding diam g max a diam g g a it has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n but the best previously known upper bound was exponential in nlogn v we give a quasipolynomial upper bound namely diam g exp o logn 4 loglogn exp loglog g o 1 for g sym n or g alt n where the implied constants are absolute this addresses a key open case of babai s conjecture on diameters of simple groups by a result of babai and seress 1992 our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

an explicit upper bound for the helfgott delta in sl 2 p abstract helfgott proved that there exists a 0 such that if s is a symmetric generating subset of sl 2 p containing 1 then either s 3 sl 2 p or s 3 s 1 it is known that 1 3024 here we show that log 2 7 1 6 0 3012 and we present evidence suggesting that this might be the true value of

growth in solvable subgroups of mathrm gl _r mathbb z p mathbb z let k mathbb z p mathbb z and let a be a subset of mathrm gl r k such that langle a rangle is solvable we reduce the study of the growth of a under the group operation to the nilpotent setting fix a positive number c ge 1 we prove that either a grows meaning a 3 ge c a or else there are groups u r and s with u r unlhd s unlhd langle a rangle such that s u r is nilpotent a k cap s is large and u r subseteq a k where k depends only on the rank r of mathrm gl r k here a k x 1 x 2 cdots x k x i in a cup a 1 cup 1 when combined with recent work by pyber and szabo the main result of this paper implies that it is possible to draw the same conclusions without supposing that langle a rangle is solvable

growth in finite simple groups of lie type we prove that if l is a finite simple group of lie type and a a set of generators of l then either a grows i e vert a 3 vert vert a vert 1 varepsilon where varepsilon depends only on the lie rank of l or a 3 l this implies that for simple groups of lie type of bounded rank a well known conjecture of babai holds i e the diameter of any cayley graph is polylogarithmic we also obtain new families of expanders a generalization of our proof yields the following let a be a finite subset of sl n mathbb f mathbb f an arbitrary field satisfying big vert a 3 big vert le mathcal k vert a vert then a can be covered by mathcal k m i e polynomially many cosets of a virtually soluble subgroup of sl n mathbb f which is normalized by a where m depends on n see more at http www ams org journals jams 0000 000 00 s0894 0347 2014 00821 3 home html sthash lp65mzge dpuf

affine linear sieve expanders and sum product let mathcal o be an orbit in n of a finitely generated subgroup of gl n whose zariski closure zcl is suitably large e g isomorphic to sl2 we develop a brun combinatorial sieve for estimating the number of points on mathcal o at which a fixed integral polynomial is prime or has few prime factors and discuss applications to classical problems including pythagorean triangles and integral apollonian packings a fundamental role is played by the expansion property of the congruence graphs that we associate with mathcal o this expansion property is established when zcl sl2 using crucially sum product theorem in q for q square free

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

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

rattle a velocity version of the shake algorithm for molecular dynamics calculations an algorithm called rattle for integrating the equations of motion in molecular dynamics calculations for molecular models with internal constraints is presented the algorithm is similar to shake which is one of the standard methods for performing such calculations rattle calculates the positions and velocities at the next time from the positions and velocities at the present time step without requiring information about the earlier history like shake it is based on the verlet algorithm and retains the simplicity of using cartesian coordinates for each of the atoms to describe the configuration of a molecule with internal constraints rattle guarantees that the coordinates and velocities of the atoms in a molecule satisfy the internal constraints at each time step rattle has two advantages over shake on computers of fixed precision it is of higher precision than shake since it deals directly with the velocities it is easier to modify rattle for use with the recently developed constant temperature and constant pressure molecular dynamics methods and with the nonequilibrium molecular dynamics methods that make use of rescaling of the atomic velocities

explicit helfgott type growth in free products and in limit groups abstract we adapt safin s result on powers of sets in free groups to obtain helfgott type growth in free products if a is any finite subset of a free product of two arbitrary groups then either a is conjugate into one of the factors or the triple product a 3 of a satisfies a 3 1 7776 a 2 or a generates an infinite cyclic or infinite dihedral group we also point out that if a is any finite subset of a limit group then a 3 satisfies the above inequality unless a generates a free abelian group this gives rise to many infinite groups g where there exist c 0 and 1 such that any finite subset a of g either satisfies a 3 c a 1 or generates a virtually nilpotent group

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 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