on g inverses and the nonsingularity of a bordered matrixabc0 abstract we consider the bordered matrix m a b c 0 with the rank additivity condition and g c 1 c 2 c 3 c 4 which is a g inverse of m we give the necessary and sufficient conditions for c 1 to be a g inverse outer inverse reflexive g inverse and moore penrose inverse of a as a special case we give necessary and sufficient conditions for m to be nonsingular and explicit expressions of c i in m 1 moreover we give a method for computing any outer inverse of a matrix

unicity of minimal rank completions for tri diagonal partial block matrices abstract in this paper we establish necessary and sufficient conditions for a minimal rank completion of a tri diagonal partial block matrix to be unique the case of 4 4 block matrices with four consecutive diagonals is also considered along with its application to the problem of completing a matrix and its inverse

solving constrained matrix equations and cramer rule this paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using cramer rule

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

on g inverses of a bordered matrix revisited abstract in this article we study the structure and properties of g inverse m of the bordered matrix m a b c 0 which is partitioned conforming with m m d 1 d 2 d 3 d 4 by applying the multiple quotient singular value decomposition qq svd from the structure of m we discuss the relations among d1 d2 d3 and d4 and derive equivalent conditions for m to be a bordered matrix by studying d1 in m we obtain some equivalent conditions for d1 to be a g inverse of a

nonnegative idempotent matrices and the minus partial order abstract we describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order

quotients of products of curves new surfaces with pg 0 and their fundamental groups we construct many new surfaces of general type with q pg 0 whose canonical model is the quotient of the product of two curves by the action of a finite group g constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type we indeed classify all such surfaces whose canonical model is singular the smooth case was classified in an earlier work as an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group g

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

grothendieck s problems concerning profinite completions and representations of groups in 1970 alexander grothendieck 6 posed the following problem let 1 and 2 be finitely presented residually finite groups and let u 1 2 be a homomorphism such that the induced map of profinite completions u 1 2 is an isomorphism does it follow that u is an isomorphism in this paper we settle this problem by exhibiting pairs of groups u p such that is a direct product of two residually finite hyperbolic groups p is a finitely presented subgroup of infinite index p is not

the nonunique parallel sum abstract this paper is a sequel to earlier study of the authors on the nonunique shorted matrix under the failure of regularity conditions when a and b are real symmetric nonnegative definite n n d matrices of the same order anderson and trapp present an alternative definition of the parallel sum of anderson and duffin which uses a shorted version of the matrix here a being real symmetric n n d the regularity conditions are trivially true in the more general case however a similar approach based on the nonunique shorted matrix leads to a nonunique parallel sum the extent to which the nonunique parallel sum retains the properties of parallel sum of anderson and duffin is examined an interesting statistical interpretation of the parallel sum operation is provided in an appendix

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

peripheral fillings of relatively hyperbolic groups in this paper a group theoretic version of dehn surgery is studied starting with an arbitrary relatively hyperbolic group g we define a peripheral filling procedure which produces quotients of g by imitating the effect of the dehn filling of a complete finite volume hyperbolic 3 manifold m on the fundamental group 1 m the main result of the paper is an algebraic counterpart of thurston s hyperbolic dehn surgery theorem we also show that peripheral subgroups of g almost have the congruence extension property and the group g is approximated in an algebraic sense by its quotients obtained by peripheral fillings

convergence of block iterative methods for linear systems arising in the numerical solution of euler equations we discuss block matrices of the forma a ij wherea ij is ak k symmetric matrix a ij is positive definite anda ij is negative semidefinite these matrices are natural block generalizations of z matrices and m matrices matrices of this type arise in the numerical solution of euler equations in fluid flow computations we discuss properties of these matrices in particular we prove convergence of block iterative methods for linear systems with such system matrices

uncountably many groups of type fp we construct uncountably many discrete groups of type fp in particular we construct groups of type fp that do not embed in any finitely presented group we compute the ordinary l 2 and compactly supported cohomology of these groups for each n at least four we construct a closed aspherical n manifold that admits an uncountable family of acyclic regular coverings with non isomorphic covering groups

matrix compensation and diagonal compensation matrix compensation is introduced to preserve the positivity of matrices when we construct preconditioners for positive definite matrices diagonal compensation for symmetric positive definite matrices is generalized to positive definite matrices if matrix compensation is used to construct preconditioners condition numbers of preconditioned matrices are estimated

finite presentations for kahler groups with arbitrary finiteness properties abstract we construct the first explicit finite presentations for a family of kahler groups with arbitrary finiteness properties answering a question of suciu

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

coxeter orbits and brauer trees abstract we study the cohomology with modular coefficients of deligne lusztig varieties associated to coxeter elements under some torsion free assumption on the cohomology we derive several results on the principal l block of a finite reductive group g f q when the order of q modulo l is assumed to be the coxeter number these results include the determination of the planar embedded brauer tree of the block as conjectured by hiss lubeck and malle 1995 in 25 and the derived equivalence predicted by the geometric version of broue s conjecture broue and malle 1993 7

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

modified tangential frequency filtering decomposition and its fourier analysis in this paper a modified tangential frequency filtering decomposition mtffd preconditioner is proposed the optimal order of the modification and the optimal relaxation parameter is determined by fourier analysis with the choice of optimal order of modification the fourier results show that the condition number of the preconditioned matrix is mathcal o h frac 2 3 and the spectrum distribution of the preconditioned matrix can be predicted by the fourier results the performance of mtffd preconditioner is compared with tangential frequency filtering tffd preconditioner on a variety of large sparse matrices arising from the discretization of pdes with discontinuous coefficients the numerical results show that the mtffd preconditioner is much more efficient than the tffd preconditioner

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

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

groups with infinite mod p schur multiplier abstract the non finiteness of the mod p schur multiplier of a finitely generated group g with trivial center implies the existence of uncountably many non residually finite non isomorphic central extension groups with kernel c p see thm a this phenomenon is related to the comparison of the cohomology of the profinite completion g of a group g and the cohomology of the group g itself see thm b according to j p serre 8 a group g is good if the cohomologies of g and g are naturally isomorphic on finite coefficients it is shown that non uniform arithmetic lattices of algebraic rank 1 groups over local fields of positive characteristic p are not good see thm c

on a class of matrices which arise in the numerical solution of euler equations we study block matricesa aij where every blocka ij k k is hermitian anda ii is positive definite we call such a matrix a generalized h matrix if its block comparison matrix is a generalized m matrix these matrices arise in the numerical solution of euler equations in fluid flow computations and in the study of invariant tori of dynamical systems we discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized h matrix

fundamental groups of links of isolated singularities 2014 american mathematical society all rights reserved we study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities we prove that for every finitely presented group g there is a complex projective surface s with simple normal crossing singularities only so that the fundamental group of s is isomorphic to g we use this to construct 3 dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to g lastly we prove that a finitely presented group g is q superperfect has vanishing rational homology in dimensions 1 and 2 if and only g if is isomorphic to the fundamental group of the link of a rational 6 dimensional complex singularity

the strong atiyah conjecture for virtually cocompact special groups we provide new conditions for the strong atiyah conjecture to lift to finite group extensions in particular we show that fundamental groups of compact special cube complexes satisfy these conditions so the conjecture holds for finite extensions of these groups

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

twisted alexander polynomials detect fibered 3 manifolds a classical result in knot theory says that for a bered knot the alexander polynomial is monic and that the degree equals twice the genus of the knot this result has been generalized by various authors to twisted alexander polynomials and bered 3 manifolds in this paper we show that the conditions on twisted alexander polynomials are not only necessary but also sucient for a 3 manifold to be bered by previous work of the authors this result implies that if a manifold of the form s 1 n 3 admits a symplectic structure then n bers over s 1 in fact we will completely determine the symplectic cone of s 1 n in terms of the bered faces of the thurston norm ball of n

random groups do not split we prove that random groups in the gromov density model at any density satisfy property fa i e they do not act non trivially on simplicial trees this implies that their gromov boundaries defined at density less than frac 1 2 are menger curves

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

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

how to express a parallel sum of k matrices abstract we present an alternative expression for the parallel sum of k hermitian nonnegative definite matrices by using the moore penrose inverse of a block matrix

on least squares g inverses and minimum norm g inverses of a bordered matrix abstract by applying the restricted singular value decomposition r svd qq svd in this article we derive structures of least squares g inverses and minimum norm g inverses of a bordered matrix m a b c 0 we also obtain the specific formulation of the pesudo inverse m

on the characteristic polynomials and eigenvalues of some specially bordered matrices abstract using a determinant identity and leverrier s algorithm for m d system we obtain the explicit expressions of the characteristic polynomial and the eigenvalues of some specially structured bordered matrices m a u v d

on parallel summability of matrices abstract it is shown that the generalized inverses characterize the parallel sum the almost positive definite a p d matrices introduced by duffin and morley 2 are of two types whose intersection is the class of quasi positive definite matrices mitra and puri 7 the a p d matrices of any one type form a saturated subclass of pairwise parallel summable a p d matrices

small cancellations over relatively hyperbolic groups and embedding theorems we generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings this generalization is then used to prove various embedding theorems for countable groups for instance we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes in particular this gives the affirmative answer to the well known question of the existence of a finitely generated group g other than 2 such that all nontrivial elements of g are conjugate

the moore penrose inverses of m n block matrices and their applications abstract this article gives the expressions for the moore penrose inverses of m n block matrices when they satisfy rank additivity conditions and presents some of their special cases and applications

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

rank invariance criterion and its application to the unified theory of least squares abstract necessary and sufficient conditions are established for the product ab c to have its rank invariant with respect to the choice of a generalized inverse b in particular cases these conditions coincide with the results of mitra they are discussed also in the statistical context of the unified theory of least squares introduced by rao

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

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

universal bounds for hyperbolic dehn surgery this paper gives a quantitative version of thurston s hyperbolic dehn surgery theorem applications include the first universal bounds on the number of nonhyperbolic dehn fillings on a cusped hyperbolic 3 manifold and estimates on the changes in volume and core geodesic length during hyperbolic dehn filling the proofs involve the construction of a family of hyperbolic conemanifold structures using infinitesimal harmonic deformations and analysis of geometric limits

on some matrix equalities for generalized inverses with applications abstract necessary and sufficient conditions are derived for the matrix equality a pn q to hold where a and n are generalized inverses of matrices some consequences and applications are also given in particular necessary and sufficient conditions are derived for the additive decompositions c a b and a a 1 a k to hold

on the wedderburn guttman theorem abstract let a be a u by v matrix of rank a and let m and n be u by g and v by g matrices respectively such that m an is nonsingular then rank a n m an 1m a a g where g rank an m an 1m a rank m an this is called wedderburn guttman theorem what happens if m an is rectangular and or singular in this paper we investigate conditions under which the regular inverse m an 1 can be replaced by a g inverse m an of some kind thereby extending the wedderburn guttman theorem the resultant conditions look similar to those arising in seemingly unrelated contexts namely cochran s and related theorems on distributions of quadratic forms involving a normal random vector

groups acting on tree graded spaces and splittings of relatively hyperbolic groups abstract tree graded spaces are generalizations of r trees they appear as asymptotic cones of groups when the cones have cut points since many questions about endomorphisms and automorphisms of groups solving equations over groups studying embeddings of a group into another group etc lead to actions of groups on the asymptotic cones it is natural to consider actions of groups on tree graded spaces we develop a theory of such actions which generalizes the well known theory of groups acting on r trees as applications of our theory we describe in particular relatively hyperbolic groups with infinite groups of outer automorphisms and co hopfian relatively hyperbolic groups

invariance properties of a triple matrix product involving generalized inverses abstract given complex valued matrices a b and c of appropriate dimensions this paper investigates certain invariance properties of the product axc with respect to the choice of x where x is a generalized inverse of b different types of generalized inverses are taken into account the purpose of the paper is three fold first to review known results scattered in the literature second to demonstrate the connection between invariance properties and the concept of extremal ranks of matrices and third to add new results related to the topic

decision problems and profinite completions of groups abstract we consider pairs of finitely presented residually finite groups p for which the induced map of profinite completions p is an isomorphism we prove that there is no algorithm that given an arbitrary such pair can determine whether or not p is isomorphic to we construct pairs for which the conjugacy problem in can be solved in quadratic time but the conjugacy problem in p is unsolvable let j be the class of super perfect groups that have a compact classifying space and no proper subgroups of finite index we prove that there does not exist an algorithm that given a finite presentation of a group and a guarantee that j can determine whether or not 1 we construct a finitely presented acyclic group h and an integer k such that there is no algorithm that can determine which k generator subgroups of h are perfect

nonarithmetic superrigid groups counterexamples to platonov s conjecture margulis showed that most arithmetic groups are superrigid platonov conjectured conversely that finitely generated linear groups which are superrigid must be of arithmetic type we construct counterexamples to platonov s conjecture

groups of units of integral group rings commensurable with direct products of free by free groups research partially supported by onderzoeksraad of vrije universiteit brussel fonds voor wetenschappelijk onderzoek belgium d g i of spain fundacion seneca of murcia cnpq and fapdf ams classification index primary 16u60 secondary 11r27 16a26

the least squares g inverse for the sum of multiple matrices in this paper we give the necessary and sufficient conditions for the set identities a 1 a 2 a k 1 3 a 1 1 3 a 2 1 3 a k 1 3 and a 1 a 2 a k 1 4 a 1 1 4 a 2 1 4 a k 1 4 which concern the least squares g inverse 1 3 inverse and the minimum norm g inverse 1 4 inverse of sum of k matrices

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

using rank formulas to characterize equalities for moore penrose inverses of matrix products a group of rank formulas related to the moore penrose inverses ab and b a are established various interesting consequences of these rank formulas are obtained the extensions of these results to the weighted moore penrose inverses ab m n and b p na m p are considered

the bianchi groups are separable on geometrically finite subgroups let d be a square free positive integer and od the ring of integers in q p id the main result of this paper is that the groups psl 2 od are

nonnegative group monotone matrices and the minus partial order bapat et al previously described a class of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus partial order in this paper we improve upon that description by first presenting a more general result that gives the precise structure of nonnegative matrices dominated by a group monotone matrix under the minus partial order as a special case we derive the complete class of nonnegative matrices dominated by a nonnegative idempotent matrix that includes the class obtained by bapat et al

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

some further remarks on the singular linear model abstract inspired by a recent article of searle 1994 we consider some specific features of estimation of x in the general linear model y x v where the model matrix x need not have full column rank and the dispersion matrix v can be singular particular attention is paid to the problems related to the invariance of some matrix expressions with respect to the choice of generalized inverses these invariance properties yield interesting matrix algebra while characterizing the best linear unbiased estimator of x

genus for groups abstract we introduce a notion of a genus g c g for a class of groups c and g c it consists of isomorphism classes of groups from c having the same profinite completion as g we show finiteness results for g c g for several important families of groups including finitely generated virtually free groups we develop formulas for the number of elements in g c g in various cases by these we find interesting examples where g c g contains only one element

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

profinite and pro p completions of poincare duality groups of dimension 3 we establish some sufficient conditions for the profinite and pro p completions of an abstract group g of type fp m resp of finite cohomological dimension of finite euler characteristic to be of type fp m over the field f p for a fixed natural prime p resp of finite cohomological p dimension of finite euler p characteristic we apply our methods for orientable poincare duality groups g of dimension 3 and show that the pro p completion p of g is a pro p poincare duality group of dimension 3 if and only if every subgroup of finite index in p has deficiency 0 and p is infinite furthermore if p is infinite but not a poincare duality pro p group then either there is a subgroup of finite index in p of arbitrary large deficiency or p is virtually zp finally we show that if every normal subgroup of finite index in g has finite abelianization and the profinite completion of g has an infinite sylow p subgroup then is a profinite poincare duality group of dimension 3 at the prime p

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

cohomology of group theoretic dehn fillings i cohen lyndon type theorems abstract this is the first paper of three papers in a row aiming to study cohomology of group theoretic dehn fillings in the present paper we prove a particular free product structure which is termed the cohen lyndon property of dehn filling kernels as an application we describe the structure of relative relation modules of dehn fillings

representations of generalized inverses of partitioned matrix involving schur complement in this article we consider some representations of 1 3 1 4 1 2 3 and 1 2 4 inverses of the partitioned matrix m which are equivalent to some rank additivity conditions we present the application of these results to generalizations of the sherman morrison woodbury type formulae

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

cohomological goodness and the profinite completion of bianchi groups the concept of cohomological goodness was introduced by j p serre in his book on galois cohomology 31 this property relates the cohomology groups of a group to those of its profinite completion we develop properties of goodness and establish goodness for certain important groups we prove for example that the bianchi groups i e the groups psl 2 o where o is the ring of integers in an imaginary quadratic number field are good as an application of our improved understanding of goodness we are able to show that certain natural central extensions of fuchsian groups are residually finite this result contrasts with examples of p deligne 5 who shows that the analogous central extensions of sp 4 z do not have this property

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests