Conjugacy and Other Properties of One-Relator Groups

In this note, we prove that certain one-relator groups are residually finite and have solvable (power) conjugacy problem, by an examination of the co-primeness of the exponent sum of some of the generators appearing in the relator.     

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.     

Engel groups I

In this series of papers we study Engel groups through their presentation, using Small Cancellation Theory. In the present paper we describe the consequences of a single Engel relator.     

K2, K3 and nilpotent ideals

In this note is given an explicit computation method to decide about the minimality of a word in the free group of rank two. A cohomological use of this method is to yield words defining one relator duality groups.     

Lebesgue relators

In this paper, the famous Lebesgue covering theorem and a useful completeness theorem ofFletcher andLindgren are extended to relator spaces, which are straightforward generalizations of Weil's uniform spaces.     

On some presentations of completely regular semigroups

In this paper we consider three types of presentations of completely regular semigroups. In each of the considered cases the solution of the word problem can be reduced to the solution of the word problem for a corresponding group presentation. As a consequence, in each of these cases the one relator presentation has a solvable word problem.     

On balanced presentations of the trivial group

To solve problems of Magnus and Stallings stated in the 1960s, we construct a balanced presentation of the trivial group such that no defining relator of this presentation could be replaced by a free generator and the group, given by the altered presentation, would still be trivial. Some other related problems on balanced presentations of the trivial group are discussed. Mathematics Subject Classification (2000) 20F05, 20F06, 57M20     

Effective Indices of Subgroups in Baumslag–Pride Groups with Free Quotients

Given a group with at least two more generators than relations, we give an effective estimate on the minimal index of a subgroup with a nonabelian free quotient. We show that the index is bounded by a polynomial in the length of the relator words. We also provide a lower bound on the index.     

Magnus subgroups of one-relator surface groups

A one-relator surface group is the quotient of an orientable surface group by the normal closure of a single relator. A Magnus subgroup is the fundamental group of a suitable incompressible sub-surface. A number of results are proved about the intersections of such subgroups and their conjugates, analogous to results of Bagherzadeh, Brodski? and Collins in classical one-relator group theory.     

The Tits alternative for non-spherical Pride groups

Pride groups, or ‘groups given by presentations in whicheach defining relator involves at most two types of generators’,include Coxeter groups, Artin groups, triangles of groups, andVinberg's groups defined by periodic paired relations. We showthat every non-spherical Pride group that is not a triangleof groups satisfies the Tits alternative.     

Solution of the membership problem of the prefix monoid in certain one-relator groups

In this work we examine to what extent small cancellation conditions imply the solution of the membership problem of the prefix monoid in one-relator groups. The main results show that with additional combinatorial conditions on the defining relator (prefix pureness and strongly cyclically reduceness), which synchronize the prefixes and an analogue of the Greendlinger Lemma for the set of prefixes (Property P( \(n\) )), the membership problem for the prefix monoid can be solved in cases not covered by previous results.     

Large groups of deficiency 1

We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator, then it is ℤ × ℤ, large or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains a ℤ × ℤ subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to free by cyclic groups, 1 relator groups and residually finite groups.     

Engel groups III

Let F denote a free group. An n-Engel word of F is an iterated commutator of the form [A, nB], where (A,B) is a pair of elements of F, which we here assume to generate a non-cyclic subgroup of F. This is the second of a series of three papers that examine some of the properties of a group defined by a suitable set of relators of the form [A _i , nB _i ], i = 1, 2, ...,m (where n is fixed). In the present paper, the techniques of [1] are generalised in a way that permits the application of the conclusions of [1] concerning the nature of the consequences of a single Engel relator to the case of a finite number of n-Engel relators, subject only to some mild hypotheses on the lengths of the relators and the value of n.     

Separating quasiconvex subgroups of right-angled Artin groups

A graph group, or right-angled Artin group, is a group given by a presentation where the only relators are commutators of the generators. A graph group presentation corresponds in a natural way to a simplicial graph, with each generator corresponding to a vertex, and each commutator relator corresponding to an edge. Suppose that G is a graph group whose corresponding graph is a tree and H is a subgroup of G. We show that if H is quasiconvex with respect to either the word metric on G or the CAT(0) metric on the universal cover of the standard complex for G, then H is separable, that is, H is the intersection of finite index subgroups of G. We also discuss some consequences relating to certain 3-manifold groups. Received: 19 July 2000; in final form: 2 March 2001 / Published online: 29 April 2002     

Decision problems for inverse monoids presented by a single sparse relator

We study a class of inverse monoids of the form M=Inv 〈X∣w=1〉, where the single relator w has a combinatorial property that we call sparse. For a sparse word w, we prove that the word problem for M is decidable. We also show that the set of words in (X∪X ⁻¹)^* that represent the identity in M is a deterministic context free language, and that the set of geodesics in the Schützenberger graph of the identity of M is a regular language.     

A note on Cayley graphs

An example is given of a finite group A of order 144, with a generating set X = {x, y} such that x³ = y² = 1 and such that the Cayley graph C(A, X) has genus 4 and characteristic −6 (both of which are small relative to the order of A), although there is no short relator of the form (xy)^r with r < 12 or of the form [x, y]^r with r < 6. Accordingly this and other possible examples do not fit into a pattern suggested by [5.], 244–268).     

Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the formS * _U T whereS andT are free inverse semigroups andU is an arbitrary finitely generated inverse subsemigroup ofS andT. We make use of recent work of Bennett to show that the word problem is decidable for any such amalgam. This is in contrast to the general situation for semigroup amalgams, where recent work of Birget, Margolis and Meakin shows that the word problem for a semigroup amalgamS * _U T is in general undecidable, even ifS andT have decidable word problem,U is a free semigroup, and the membership problem forU inS andT is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for theD-classes of such an amalgam to be finite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has finiteD-classes and is combinatorial. We also obtain information concerning when such an amalgam isE-unitary: for example every one relator amalgam of the formInv<A ∪B :u =v > whereA andB are disjoint andu (resp.v) is a cyclically reduced word overA ∪A ⁻¹ (resp.B ∪B ⁻¹) isE-unitary. Research of all authors supported by a grant from the Italian CNR. The first and third authors’ research was partially supported by MURST. The second author’s research was also partially supported by NSF and the Center for Communication and Information Science of the University of Nebraska at Lincoln.     

Exactness of one relator groups

A discrete group is -exact if the reduced crossed product with converts a short exact sequence of --algebras into a short exact sequence of -algebras. A one relator group is a discrete group admitting a presentation where is a countable set and is a single word over . In this short paper we prove that all one relator discrete groups are -exact. Using the Bass-Serre theory we also prove that a countable discrete group acting without inversion on a tree is -exact if the vertex stabilizers of the action are -exact.

     

隐马尔可夫模型在语音识别中的应用

隐马尔可夫模型 (HMM)的基本技术是语音识别中较为成功的算法 .主要是它具有较强的对时间序列结构的建模能力 .本文首先深入浅出地介绍了 HMM的基本技术和一个基于 HMM的孤立词语音识别系统的构成方法 ,其次 ,基于 HMM尚存有一些缺陷 ,造成语音识别能力较弱 ,为此本文又进一步阐述了语音识别应用中的几种改进的 HMM系统及目前的热点方法—— HMM与 ANN构成的混合网络     

An investigation on ordered algebraic hyperstructures

In this paper, we present some basic notions of simple ordered semihypergroups and regular ordered Krasner hyperrings and prove some results in this respect. In addition, we describe pure hyperideals of ordered Krasner hyperrings and investigate some properties of them. Finally, some results concerning purely prime hyperideals are proved.