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

the isometry group of outer space abstract we prove analogues of royden s theorem for the lipschitz metrics of outer space namely that isom cv n out f n

mapping tori of free group automorphisms are coherent the mapping torus of an endomorphism of a group g is the hnnextensiong g with bonding maps the identity and we show that a mapping torus of an injective free group endomorphism has the property that its flnitely generated subgroups are flnitely presented and moreover these subgroups are of flnite type

connectedness properties of limit sets we study convergence group actions on continua and give a criterion which ensures that every global cut point is a parabolic fixed point we apply this result to the case of boundaries of relatively hyperbolic groups and consider implications for connectedness properties of such spaces

meromorphic quadratic differentials and measured foliations on a riemann surface we describe the space of measured foliations induced on a compact riemann surface by meromorphic quadratic differentials we prove that any such foliation is realized by a unique such differential q if we prescribe in addition the principal parts of sqrt q at the poles this generalizes a theorem of hubbard and masur for holomorphic quadratic differentials the proof analyzes infinite energy harmonic maps from the riemann surface to mathbb r trees of infinite co diameter with prescribed behavior at the poles

doubles finiteness properties of groups and quadratic isoperimetric inequalities abstract we describe a doubling construction that gives many new examples of groups that satisfy a quadratic isoperimetric inequality using this construction we prove that the presence of a quadratic isoperimetric inequality does not constrain the higher finiteness properties of a group in contrast to the sub quadratic case

on the positive solutions of the difference equation xn 1 xn 1 1 x nxn 1 the positive solutions of the difference equation x n 1 x n 1 1 x nx n 1 n 0 1 2 are investigated where x 1 and x 0 are the positive real numbers

cubulating graphs of free groups with cyclic edge groups we prove that if g is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups and g has no non euclidean baumslag solitar subgroups then g is the fundamental group of a compact nonpositively curved cube complex in addition if g is also word hyperbolic i e if g contains no baumslag solitar subgroups of any type we show that g is linear in fact is a subgroup of sl n z

global behavior of a higher order difference equation in this paper we study the global behavior of the nonnegative equilibrium points of the difference equation x n 1 ax n email protected i 02k 1x n i n 0 1 where a b c are nonnegative parameters initial conditions are nonnegative real numbers and k m are nonnegative integers email protected 2k 1 also we derive solutions of some special cases of this equation

cannon thurston maps for trees of hyperbolic metric spaces let x d be a tree t of hyperbolic metric spaces satisfying the quasi isometrically embedded condition let v be a vertex of t let x v d v denote the hyperbolic metric space corresponding to v then i x v x extends continuously to a map i x v x this generalizes a theorem of cannon and thurston the techniques are used to give a new proof of a result of minsky thurston s ending lamination conjecture for certain kleinian groups applications to graphs of hyperbolic groups and local connectivity of limit sets of kleinian groups are also given

an upper bound on reidemeister moves we provide an explicit upper bound on the number of reidemeister moves required to pass between two diagrams of the same link this leads to a conceptually simple solution to the equivalence problem for links

genericity the arzhantseva ol shanskii method and the isomorphism problem for one relator groups we show that the isomorphism problem is solvable in at most exponential time for a class of one relator groups which is exponentially generic in the sense of ol shanskii this is obtained by applying the arzhantseva ol shanskii graph minimization method to prove the general result that for fixed integers m 2 and n 1 there is an exponentially generic class of non free m generator n relator groups with the property that there is only one nielsen equivalence class of m tuples which generate a non free subgroup in particular every m generated subgroup in such a generic group g is either free or is equal to g itself and such groups are thus co hopfian these results are obtained by elementary methods without using the deep results of sela about co hopficity and the isomorphism problem for torsion free hyperbolic groups

homogeneity in the free group we show that any nonabelian free group f is strongly 0 homogeneous that is that finite tuples of elements which satisfy the same first order properties are in the same orbit under aut f we give a characterization of elements in finitely generated groups which have the same first order properties as a primitive element of the free group we deduce as a consequence that most hyperbolic surface groups are not strongly 0 homogeneous 1 introduction since the works of sela sel1 sel5 and kharlampovich and myasnikov km on tarski s problem which showed that finitely generated free groups of rank at least 2 all have the same first order theory there has been renewed interest in model theoretic questions about free groups with sela s work it has become clear that techniques of geometric group theory provide extremely effective ways to tackle these questions this can be seen in subsequent results such as sel5 where sela showed that the theory of the free group is stable or in per2 where the elementary subgroups of a free group of finite rank are shown to be exactly its nonabelian free factors moreover the geometric nature of the tools often allows these results to be generalized to the class of torsion free hyperbolic groups see sel4 in this paper we apply these techniques to study types of elements and homogeneity in torsion free hyperbolic groups particularly in free groups and hyperbolic surface groups the type of a tuple n a in a structure is the set of all first order formulas it realizes more formally let l be a first order language let m be an l structure and let b be a subset of m for basic definitions of model theory the reader is referred to ck mar or the short survey given in cha if n a is a k tuple of elements of m the type tp m n a b of n a over b in m is the set of all formulas with k free variables and parameters in b which n a realizes in m an l structure m is strongly 0 homogeneous if for any pair of finite tuples n a n a 0 with tp m n a d tp m n a 0 there is an automorphism of m sending n a to n a 0

the equivalence of some residual properties of word hyperbolic groups abstract we show that every word hyperbolic group is residually finite if and only if every word hyperbolic group has a finite quotient

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

moment of cepstrum and its applications it is shown that the moments of a signal sequence and of its corresponding cepstral sequence are related in a way which obviates the need for the direct calculation of the cepstral coefficients hence an ideal calculation for the moments of the cepstrum is possible even if the duration of the cepstrum is infinite it is also possible to describe all the properties of a signal sequence from its moments the effectiveness of this method is demonstrated in the problems of homomorphic deconvolution and echo removal it is also shown that in linear filtering problems the moments of the minimum phase sequence are calculated from the corresponding linear phase sequence without actually calculating the minimum phase sequence for a discrete time real signal sequence with finite duration complete equivalence is demonstrated between the signal sequence and its moments this permits a new description and a new set of design specifications based on moments for such systems

conjugacy separability of 1 acylindrical graphs of free groups we use the theory of group actions on profinite trees to prove that the fundamental group of a finite 1 acylindrical graph of free groups with finitely generated edge groups is conjugacy separable this has several applications we prove that positive c 1 6 one relator groups are conjugacy separable we provide a conjugacy separable version of the rips construction we use this latter to provide an example of two finitely presented residually finite groups that have isomorphic profinite completions such that one is conjugacy separable and the other does not even have solvable conjugacy problem

polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms we prove that every mapping torus of any free group endomorphism is residually finite we show how to use a not yet published result of e hrushovski to extend our result to arbitrary linear groups the proof uses algebraic self maps of affine spaces over finite fields in particular we prove that when such a map is dominant the set of its fixed closed scheme points is zariski dense in the affine space

the residual finiteness of negatively curved polygons of finite groups a subgroup m g is almost malnormal provided that for each g g m the intersection m g m is finite it is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup is residually finite a consequence of a generalization of this result is that an acute angled n gon of finite groups is residually finite if n 4 another consequence is that if g acts properly discontinuously and cocompactly on a 2 dimensional hyperbolic building whose chambers have acute angles and at least 4 sides then g is residually finite

canonical representatives and equations in hyperbolic groups we use canonical representatives in hyperbolic groups to reduce the theory of equations in torsion free hyperbolic groups to the theory in free groups as a result we get an effective procedure to decide if a system of equations in such groups has a solution for free groups this question was solved by makanin ma and razborov ra the case of quadratic equations in hyperbolic groups has already been solved by lysenok ly our whole construction plays an essential role in the solution of the isomorphism problem for torsion free hyperbolic groups se1 se2

direct nonrecursive relations between cepstrum and predictor coefficients direct i e nonrecursive relations are derived for the cepstrum in terms of the predictor coefficients and vice versa connections with algebraic roots symmetric functions statistical moments and cumulants are pointed out some implications for pitch detection are also discussed

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

the conjugacy and isomorphism problems for combable groups there exist combable groups in which the conjugacy problem is unsolvable the isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups

calculating the complex cepstrum without phase unwrapping or integration the complex cepstrum as developed by oppenheim 4 and oppenheim schafer and stockham 5 originated from the pioneering paper of bogert healy and tukey 3 it is a nonlinear transformation from one signal space into another which converts two superposed signals into their sum the cepstrum is usually computed by inverse fourier transformation of the frequency domain complex logarithm of the original signal the phase term or imaginary part of this complex logarithm presents a computational difficulty which is sometimes approached by integration of the derivative of the phase spectrum an excellent approach to phase computation is the adaptive integration methodology as proposed by tribolet 10 unfortunately this technique can be computationally intensive this note describes a simple technique for efficient computation of the complex cepstrum which requires no phase unwrapping or integration the procedure is given in both one and two dimensions

subgroup properties of fully residually free groups we prove that fully residually free groups have the howson property that is the intersection of any two finitely generated subgroups in such a group is again finitely generated we also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup

on the positive solutions of the difference equation xn 1 axn 11 bxnxn 1 we study the positive solutions of the difference equation x n 1 ax n 11 bx nx n 1 n 0 1 2 where a b x 1 and x 0 are non negative real numbers

recognizing a relatively hyperbolic group by its dehn fillings dehn fillings for relatively hyperbolic groups generalize the topological dehn surgery on a non compact hyperbolic 3 manifold such as a hyperbolic knot complement we prove a rigidity result saying that if two non elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic dehn fillings then these groups are in fact isomorphic our main application is a solution to the isomorphism problem in the class of non elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings

dynamics of a rational difference equation abstract in this note we investigate the solution of the difference equation x n 1 x n 1 a x n 1 x n n 0 1 2 where x 1 x 0 r and a 0 moreover we discuss the stability properties and semi cycle behavior of this solution

discrete and free two generated subgroups of sl2 over non archimedean local fields abstract we present a practical algorithm which given a non archimedean local field k and any two elements a b s l 2 k determines after finitely many steps whether or not the subgroup a b s l 2 k is discrete and free of rank two this makes use of the ping pong lemma applied to the action of s l 2 k by isometries on its bruhat tits tree the algorithm itself can also be used for two generated subgroups of the isometry group of any locally finite simplicial tree and has applications to the constructive membership problem in an appendix joint with frederic paulin we give an erratum to his 1989 paper the gromov topology on r trees which details some translation length formulae that are fundamental to the algorithm

solvable groups with polynomial dehn functions given a finitely presented group h finitely generated subgroup b of h and a monomorphism b h we obtain an upper bound of the dehn function of the corresponding hnn extension g hh t j t i1 bt b i in terms of the dehn function of h and the distortion of b in g using such a bound we construct first examples of non polycyclic solvable groups with polynomial dehn functions the constructed groups are metabelian and contain the solvable baumslag solitar groups in particular this answers a question posed by birget ol shanskii rips and sapir

virtual betti numbers of mapping tori of 3 manifolds given a reducible 3 manifold m with an aspherical summand in its prime decomposition and a homeomorphism f m rightarrow m we construct a map of degree one from a finite cover of m rtimes f s 1 to a mapping torus of a certain aspherical 3 manifold we deduce that m rtimes f s 1 has virtually infinite first betti number except when all aspherical summands of m are virtual t 2 bundles this verifies all cases of a conjecture of t j li and y ni that any mapping torus of a reducible 3 manifold m not covered by s 2 times s 1 has virtually infinite first betti number except when m is virtually n t 2 rtimes s 1 ms 2 times s 1 li ni s conjecture was recently confirmed by ni with a group theoretic result namely by showing that there exists a pi 1 surjection from a finite cover of any mapping torus of a reducible 3 manifold to a certain mapping torus of m s 2 times s 1 and using the fact that free by cyclic groups are large when the free group is generated by more than one element

cubulating hyperbolic free by cyclic groups the irreducible case let v be a nite graph and let v v be an irreducible train track map whose mapping torus has word hyperbolic fundamental group g then g acts freely and cocompactly on a cat 0 cube complex hence if f is a nite rank free group and f f an irreducible monomorphism so that g f is word hyperbolic then g acts freely and cocompactly on a cat 0 cube complex this holds in particular if is an irreducible automorphism with g f o z word hyperbolic

infinitely many hyperbolic 3 manifolds which contain no reebless foliation calegari ca has shown that if m is also atoroidal then or1 m is gromov negatively curved furthermore thurston has proposed an approach to demonstrating geometrization for such m many 3 manifolds contain reebless foliations and it has often been conjectured that all closed hyperbolic 3 manifolds do it is our impression that for many years hatcher provided the sole voice of dissent in this paper we give the first examples of closed hyperbolic 3 manifolds which contain no reebless foliation

virtual betti numbers and virtual symplecticity of 4 dimensional mapping tori in this note we compute the virtual first betti numbers of 4 manifolds fibering over s 1 with prime fiber as an application we show that if such a manifold is symplectic with nonpositive kodaira dimension then the fiber itself is a sphere or torus bundle over s 1 in a different direction we prove that if the 3 dimensional fiber of such a 4 manifold is virtually fibered then the 4 manifold is virtually symplectic unless its virtual first betti number is 1

scalar lpc quantization based on format jnd s efficient scalar quantization tables for lpc k parameters were developed using a distortion measure based on just noticeable differences jnd s in formant parameters of the speech spectrum envelope forty percent fewer bits were required than the 41 frame used in conventional approaches an empirical technique was developed for relating perturbations in k parameters and formant parameters new estimates were obtained for the values of the formant jnd s they are about four times the steady state values reported by flanagan 6 and increase sharply above approximately 1 5 khz

convergence of freely decomposable kleinian groups we consider a compact orientable hyperbolic 3 manifold with a compressible boundary suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination we prove that if this limit projective lamination is doubly incompressible then the sequence has compact closure in the deformation space as a consequence we generalise thurston s double limit theorem and solve his conjecture on convergence of function groups affirmatively

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

the word problem in the baumslag group with a non elementary dehn function is polynomial time decidable abstract we prove that the word problem in the baumslag group g 1 2 a b a a b a 2 which has a non elementary dehn function is decidable in polynomial time

deciding isomorphy using dehn fillings the splitting case we solve dehn s isomorphism problem for virtually torsion free relatively hyperbolic groups with nilpotent parabolic subgroups we do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups namely the isomorphism problem separation of torsion in their outer automorphism groups by congruences and the mixed whitehead problem an automorphism group orbit problem the first step of the reduction is to compute canonical jsj decompositions dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and if so whether a global isomorphism can be assembled for the class of finitely generated nilpotent groups we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups

on the difference equation xn 1 xn 1 1 xnxn 1 in this paper we investigate the solutions of the difference equation x n 1 x n 1 1 x nx n 1 n 0 1 2 where x 1 and x 0 are the real numbers such that x 1x 0 1

quasi isometrically embedded subgroups of braid and diffeomorphism groups we show that a large class of right angled artin groups in particular those with planar complementary defining graph can be embedded quasi isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary with respect to the norm metric this extends results of benaim and gambaudo who gave quasi isometric embeddings of and for all as a consequence we are also able to embed a variety of gromov hyperbolic groups quasi isometrically in pure braid groups and in the group examples include hyperbolic surface groups some hnn extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3 manifold

on the hyperbolicity of small cancellation groups and one relator groups in the article a result relating to maps finite planar connected and simply connected 2 complexes that satisfy a c p t it is shown that a finitely generated onerelator group g whose reduced relator r is of the form r at0at1 atn 1 where the words ti are distinct and have no occurrences of the letter a 1 is not hyperbolic if and only if one has in the free group that 1 n 2 and t0t 1 1 is a proper power 2 n 3 and for some i it is true with subscripts mod 3 that tit 1 i 1tit 1 i 2 1 3 n 4 and for some i it is true with subscripts mod 4 that tit 1 i 1ti 2t 1 i 3 1

the cannon thurston map for punctured surface groups let be the fundamental group of a compact surface group with non empty boundary we suppose that admits a properly discontinuous strictly type preserving action on hyperbolic 3 space such that there is a positive lower bound on the translation lengths of loxodromic elements we describe the cannon thurston map in this case in particular we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3 space where the action on the circle is obtained by taking any finite area complete hyperbolic structure on the surface and lifting to the boundary of hyperbolic 2 space we deduce that the limit set is locally connected hence a dentrite in the singly degenerate case moreover we show that the cannon thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3 manifold for closed surface bundles over the circle this was obtained by cannon and thurston some generalisations and variations have been obtained by minsky mitra alperin dicks porti mcmullen and cannon we deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected

time domain cepstral transformations the realization of homomorphic systems for convolution is addressed motivated by the limitations of the fourier transform based method commonly used in homomorphic complex cepstral filtering the time domain cepstral transformation tdct method which is entirely based on time domain calculations thus avoiding or minimizing the problems associated with the ft method is introduced explicit transformations of an ordinary mixed phase time sequence belonging to convolution space into its complex cepstrum time sequence belonging to additive cepstrum vector space and vice versa are derived the method does not require unwrapped phase calculations and no specific windows are used to precondition the signal in order to produce a more accurate representation of the complex cepstrum it trades reduced computational efficiency for improved performance examples comparing fourier based and td cepstrum transformations are presented

the simplicial volume of mapping tori of 3 manifolds we prove that any mapping torus of a closed 3 manifold has zero simplicial volume when the fiber is a prime 3 manifold classification results can be applied to show vanishing of the simplicial volume however the case of reducible fibers is by far more subtle we thus analyse the possible self homeomorphisms of reducible 3 manifolds and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori to this end we introduce a new technique for understanding self homeomorphisms of connected sums in arbitrary dimensions on the level of classifying spaces and for computing the simplicial volume in particular we extend our computations to mapping tori of certain connected sums in higher dimensions our main result completes the picture for the vanishing of the simplicial volume of fiber bundles in dimension four moreover we deduce that dimension four together with the trivial case of dimension two are the only dimensions where all mapping tori have vanishing simplicial volume as a group theoretic consequence we derive an alternative proof of the fact that the fundamental group g of a mapping torus of a 3 manifold m is gromov hyperbolic if and only if m is virtually a connected sum s 2 times s 1 and g does not contain mathbb z 2

acylindrical accessibility for groups acting on ℝ trees we prove an acylindrical accessibility theorem for finitely generated groups acting on trees namely we show that if g is a freely indecomposable non cyclic k generated group acting minimally and d acylindrically on an tree x then there is a finite subtree open image in new window of measure at most 2d k 1 such that gt x this generalizes theorems of z sela and t delzant about actions on simplicial trees

on suspensions and conjugacy of hyperbolic automorphisms we remark that the conjugacy problem for pairs of hyperbolic automorphisms of a finitely presented group typically a free group is decidable the solution that we propose uses the isomorphism problem for the suspensions and the study of their automorphism group

doubles of groups and hyperbolic lerf 3 manifolds we show that the quasiconvex subgroups in doubles of certain negatively curved groups are closed in the profinite topology this allows us to construct the first known large family of hyperbolic 3 manifolds such that any finitely generated subgroup of the fundamental group of any member of the family is closed in the profinite topology

on the asymptotic stability of xn 1 a xnxn k xn xn k we prove that the equilibrium solution of the rational difference equation x n 1 a x nx n kx n x n k n 0 1 2 where k is a nonnegative integer a 0 and x k x 0 0 is globally asymptotically stable