stallings foldings and subgroups of free groups we re cast in a more combinatorial and computational form the topological approach of j stallings to the study of subgroups of free groups

normalisers in limit groups let gamma be a limit group s subset gamma a non trivial subgroup and n the normaliser of s if h 1 s mathbb q has finite mathbb q dimension then s is finitely generated and either n s is finite or n is abelian this result has applications to the study of subdirect products of limit groups

on pro p analogues of limit groups via extensions of centralizers we begin a study of a pro p analogue of limit groups via extensions of centralizers and call mathcal l this new class of pro p groups we show that the pro p groups of mathcal l have finite cohomological dimension type fp and non positive euler characteristic among the group theoretic properties it is proved that they are free by torsion free nilpotent and if non abelian do not have a finitely generated non trivial normal subgroup of infinite index furthermore it is shown that every 2 generated pro p group in the class mathcal l is either free pro p or abelian

subgroup separability in residually free groups we prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type mathrm fp infty are virtual retracts we describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups

a numerical approach for solving an extended fisher kolomogrov petrovskii piskunov equation in the present paper a numerical method based on finite differences and spline collocation is presented for the numerical solution of a generalized fisher integro differential equation a composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed form difference scheme a number of test examples are solved to assess the accuracy of the method the numerical solutions obtained indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods convergence and stability of the scheme have also been discussed

transient energy flow between two coupled beams abstract two coupled beams were excited with an impulse at one end of the source beam and the response monitored at the remote end of the receiver beam the response was predicted with an exact wave solution and compared with two approximate solutions a first wave prediction and a transient statistical energy solution with a numerical parameter study the peak velocity peak energy and rise time to peak energy were compared with modal overlap used as the independent variable the applicability of these two approximate methods was defined in terms of modal overlap the approximate methods were found to be equally accurate for both longitudinal and flexural waves

on the finite presentation of subdirect products and the nature of residually free groups we establish it virtual surjection to pairs vsp as a general criterion for the finite presentability of subdirect products of groups if gamma 1 ldots gamma n are finitely presented and s we use the vsp criterion to characterize the finitely presented residually free groups we prove that the class of such groups is recursively enumerable we describe an algorithm that given a finite presentation of a residually free group constructs a canonical embedding into a direct product of finitely many limit groups we solve the multiple conjugacy problem and membership problem for finitely presentable subgroups of residually free groups we also prove that there is an algorithm that given a finite generating set for such a subgroup will construct a finite presentation new families of subdirect products of free groups are constructed including the first examples of finitely presented subgroups that are neither rm fp infty nor of stallings bieri type

implicit function theorem over free groups abstract we introduce the notion of a regular quadratic equation and a regular ntq system over a free group we prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and ntq systems we will also show that the implicit function theorem is true only for these varieties in algebraic geometry such results would be described as lifting solutions of equations into generic points from the model theoretic view point we claim the existence of simple skolem functions for particular formulas over free groups proving these theorems we describe in details a new version of the makanin razborov process for solving equations in free groups we also prove a weak version of the implicit function theorem for ntq systems which is one of the key results in the solution of the tarski s problems about the elementary theory of a free group we call it the parametrization theorem

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

a hybrid finite element statistical energy analysis method for impulsive and transient loading abstract the hybrid approach coupling statistical energy analysis sea and the finite element method has become a prominent technique for analysing structures under steady state loads in the mid frequency range where some components behave in a deterministic manner with low modal density and others in a statistical manner with high modal density and statistical overlap in this paper the method is extended from its current steady state capability to provide calculation of structural responses under impulsive and time varying loads similar to the steady state method a system is split into deterministic components that are modelled using the finite element approach and statistical components that are modelled as sea subsystems an evolutionary spectrum approach based on the priestley description of random processes is applied to model the response of both the sea and deterministic components which are coupled by considering a power balance between the sea subsystems the diffuse field reciprocity relationship that relates the reverberant forces generated by a subsystem to the energy within it is explored under transient conditions where it is found that it can be important to account for the build up of a reverberant field following an impulse results from the method are compared against finite element simulations for a system of plates coupled by a beam and it is found to generate predictions with the accuracy expected of an sea based method although deterministic oscillations of the deterministic system at early times are not accounted for

irreducible affine varieties over a free group ii systems in triangular quasi quadratic form and description of residually free groups abstract we shall prove the conjecture of myasnikov and remeslennikov 4 which states that a finitely generated group is fully residually free every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group if and only if it is embeddable in the lyndon s exponential groupfz x which is thez x completion of the free group herez x is the ring of polynomials of one variable with integer coefficients historically lyndon s attempts to solve tarski s famous problem concerning the elementary equivalence of free groups of different ranks led him to introducefz x an free group is a groupgsuch that the class of formulas true ing is the same as the class of formulas true in a nonabelian free group a finitely generated group is free if and only if it is fully residually free 22 our result gives an algebraic description of free groups we shall give an algorithm to represent a solution set of an arbitrary system of equations overfas a union of finite number of irreducible components in the zariski topology onfn the solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms the possibility of such a decomposition for a solution set was conjectured by razborov in 20 and also by rips we shall give a description of systems of equations determining irreducible components using methods developed in 13 19 it is possible to find some of these methods in 18 we are thankful to e rips for attracting our attention to these techniques

subgroups of direct products of limit groups if ti tn are limit groups and s c t x x tn is of type fp q then s contains a subgroup of finite index that is itself a direct product of at most n limit groups this answers a question of sela

cubulating malnormal amalgams let g be a group splitting as a graph of groups where each vertex group acts properly on a cat 0 cube complex when g is hyperbolic relative to virtually abelian subgroups we examine conditions producing a proper action of g on a cat 0 cube complex when g is hyperbolic this action is also cocompact

automorphisms of the category of free lie algebras we prove that every automorphism of the category of free lie algebras is a semi inner automorphism this solves problem 3 9 from g mashevitzky b plotkin e plotkin electron res announc amer math soc 8 2002 1 10 for lie algebras

elementary theory of free non abelian groups we prove that any two non abelian free groups have the same elementary theory and that this theory is decidable these results solve two questions that were raised by tarski in 1945

krull dimension of solvable groups in this paper we prove that free solvable groups have finite krull dimension in fact this is true for much wider class of solvable groups termed rigid groups along the way we study the algebraic structure of the limit solvable groups fully residually free solvable groups

quasi potency and cyclic subgroup separability we study the profinite topology on discrete groups and in particular the property of cyclic subgroup separability we investigate the class of quasi potent cyclic subgroup separable groups producing many examples and showing how it behaves with respect to certain group constructions

finding all solutions of equations in free groups and monoids with involution this paper presents a pspace algorithm which yields a finite graph of exponential size that describes the set of all solutions of equations in free groups as well as the set of all solutions of equations with rational constraints in free monoids this became possible due to the recent recompression technique while this technique was successfully applied for pure word equations without involution or rational constraints it could not be used as a black box for free groups actually the presence of an involution and rational constraints complicates the situation and some additional analysis is necessary still the technique is general enough to accommodate both extensions in the end it simplifies proofs that satisfiability of word equations is in pspace and the corresponding result for equations in free groups with rational constraints as a byproduct we can decide in pspace whether the solution set is finite

complexity volumes of splittable groups abstract using graph of groups decompositions of finitely generated groups we define euler characteristic type invariants which are non zero in many interesting classes of hyperbolic limit and csa groups including elementarily free groups and one ended torsion free hyperbolic groups whose jsj decomposition contains a maximal hanging fuchsian vertex group

a derivation of the transient statistical energy analysis tsea equations with benchmark applications to plate systems abstract statistical energy analysis sea is a well established method for predicting the response of complex systems to high frequency steady state harmonic or random excitation the method has also been applied to transient and shock loading in which case it is referred to as transient sea tsea although the validity of the approach is less certain for this case since the tsea equations mix time and frequency descriptions of the response in an ad hoc way in this paper the tsea equations are derived in a new way by employing an analogy of the priestley description of a non stationary random process a key feature is that shock loading is deterministic so that the random ensemble of responses arises from random structural properties rather than random loading and this requires a reinterpretation of the priestley description the present derivation of tsea enables the appropriate initial conditions on the equations to be established and bounds on the prediction error are found from parseval s theorem the derived equations are applied to numerical and experimental examples involving plates

algebraic geometry over groups i algebraic sets and ideal theory abstract the object of this paper which is the first in a series of three is to lay the foundations of the theory of ideals and algebraic sets over groups

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

on systems of equations over free products of groups using an analogue of the makanin razborov diagrams we give a description of the solution set of systems of equations over an equationally noetherian free product of groups g equivalently we give a parametrisation of the set hom h g of all homomorphisms from a finitely generated group h to g furthermore we show that every algebraic set over g can be decomposed as a union of finitely many images of algebraic sets of ntq systems if the universal horn theory of g the theory of quasi identities is decidable then our constructions are effective 2010 elsevier inc

volume gradients and homology in towers of residually free groups we study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups g n g of increasing finite index in a fixed finitely generated group g assuming bigcap n g n 1 we focus in particular on finitely presented residually free groups calculating their ell 2 betti numbers rank gradient and asymptotic deficiency if g is a limit group and k is any field then for all j ge 1 the limit of dim h j g n k g g n as n rightarrow infty exists and is zero except for j 1 where it equals chi g we prove a homotopical version of this theorem in which the dimension of dim h j g n k is replaced by the minimal number of j cells in a k g n 1 this includes a calculation of the rank gradient and the asymptotic deficiency of g both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of slow groups we prove that if a residually free group g is of type mathrm fp m but not of type mathrm fp infty then there exists an exhausting filtration by normal subgroups of finite index g n so that lim n dim h j g n k g g n 0 hbox for j le m if g is of type mathrm fp infty then the limit exists in all dimensions and we calculate it

the equation xp yq zr and groups that act freely on l trees let g be a finitely generated group that acts freely on a lambda tree where lambda is an ordered abelian group and let x y z be elements in g we show that if x p y q z r with integers p q r geq 4 then x y and z commute