A deterministic algorithm to decide if a finitely presented abelian monoid is cancellative

In this paper we give an algorithm to compute a finite presentation for any finitely generated commutative cancellative monoid, and in particular we apply it to derive an algorithm to decide whether a finitely presented commutative monoid is cancellative or not.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

Strongly taut finitely generated monoids

A strongly taut monoid is a monoid in which all the powers of any element of the monoid have the same elasticity, that is, the ratio between the maximum and the minimum length of the factorizations of an element remains unchanged under powers. We give a procedure to determine if a finitely generated commutative monoid is strongly taut.     

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.     

Note on residually finite rings

This paper is on the subject of residually finite (= RF) modules and rings introduced by Varadarajan [93] and [98/99]. Specifically there are several theorems that simplify proofs and generalize some results of Varadarajan, namely.

Theorem 1. An RF right R-module is finitely bedded (= has finite essential socle iff M is finite.

Corollay. If T is a right RF woth just finitely many simple ringht R-modules, them R is fimite.

Theorem 2. A commutative ring R is residually finite iff every local ring R_m at a maximal ideal m is finite.     

Irreducible actions

Subdirectly irreducible acts and finitely subdirectly irreducible acts are characterized, for acts over a commutative semigroup that satisfy finiteness conditions.      

Euler Characteristics on virtually free products

We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These formulas are obtained from a general result which applies to the rank gradient and the first L²?Betti number of a finitely generated group.     

On residually finite groups of finite general rank

Following A. I.Mal’tsev, we say that a group G has finite general rank if there is a positive integer r such that every finite set of elements of G is contained in some r-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similarMal’tsev result on the Hopf property of every finitely generated residually finite group.     

An introduction to presentations of monoid acts: quotients and subacts

The purpose of this article is to introduce the theory of presentations of monoids acts. We aim to construct “nice” general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M-act A and a subact B of A, on the one hand, we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand, we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has a finite complement in an M-act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.     

Fundamental groups,inverse schützenberger automata,and monoid presentations

This paper gives decidable conditions for when a finitely generated subgroup of a free group is the fundamental group of a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Also, generalizations are given to specific types of inverse monoids as well as to monoids which are "nearly inverse." This result has applications to computing membership for inverse monoids in a Mal'cev product of the pseudovariety of semilattices with a pseudovariety of groups.

This paper also shows that there is a bijection between strongly connected inverse automata and subgroups of a free group, generated by positive words. Hence, we also obtain that it is decidable whether a finite strongly connected inverse automaton is a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Again, we have generalizations to other types of inverse monoids and to "nearly inverse" monoids. We show that it is undecidable whether a finite strongly connected inverse automaton is a Schützenberger automaton of a monoid presentation of anE-unitary inverse monoid.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.     

Periodic subsemigroups of endomorphism monoids

If S is a periodic subsemigroup of the endomorphism monoid of a polycyclic group, then Endimioni (Mediterr J Math 8:307–313, 2011) proved that S is locally finite. Here we present an alternative proof that also extends the result to groups with suitable rank restrictions. Further we give an alternative proof of McNaughton and Zalcstein’s (J Algebra 34:292–299, 1975) theorem that periodic multiplicative subsemigroups of a matrix ring over a field are also locally finite. Finally we extend the latter to periodic subsemigroups of the endomorphism ring of a finitely generated module over a commutative ring.     

Amenable actions of nonamenable groups

We present two methods of constructing amenable (in the sense of Greenleaf) actions of nonamenable groups. In the first part of the paper, we construct a class of faithful transitive amenable actions of the free group using Schreier graphs. In the second part, we show that every finitely generated residually finite group can be embedded into a bigger residually finite group, which acts level-transitively on a locally finite rooted tree, so that the induced action on the boundary of the tree is amenable on every orbit. Bibliography: 25 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 85–96.     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

On the number of generators of a projective module

In this article we give a bound on the number of generators of a finitely generated projective module of constant rank over a commutative Noetherian ring in terms of the rank of the module and the dimension of the ring. Under certain conditions we provide an improvement to the Forster–Swan bound in case of finitely generated projective modules of rank n over an affine algebra over a finite field or an algebraically closed field.     

Limits of Pure-Injective Cotilting Modules

The set of pure-injective cotilting modules over an artin algebra is shown to have a monoid structure. This monoid structure does not restrict down to a monoid structure on the finitely generated cotilting modules in general, but it does whenever the algebra is of finite representation type. Pure-injective cotilting modules are also constructed from any set of finitely generated cotilting modules with bounded injective dimension. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G10, 16P20, 16E30.     

The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every positive one-relator group is residually finite. It is shown that positive one-relator groups are generically and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs. Received: August 4, 2000     

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.