Showing papers in "International Journal of Algebra and Computation in 2001"

Closed subgroups in pro-v topologies and the extension problem for inverse automata

Abstract: We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiĭ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups. Resume: Nous etablissons un lien entre le probleme du calcul de l'adheerence d'un sous-groupe finiment engendre du groupe libre dans la topologie pro-V, ou V est une pseudovariete de groupes finis, et un probleme d'extension pour les automates inversifs qui peut etre enonce de la fac con suivante: etant donnees des transformations partielles injectives d'un ensemble fini, peuvent-elles etre etendues en des permutations qui engendrent un groupe dans V? Les deux problemes sont equivalents si V est fermee par extensions. Nous interessant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adherence pro-p d'un sous-groupe finiment engendre du groupe libre en temps polynomial et pour calculer effectivement sa cloture pro-nilpotente. Enfin nous appliquons nos resultats a un probleme de theorie des monoides finis, celui de de l'appartenance dans les pseudovarietes de monoides inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariete de groupes.

Length and area functions on groups and quasi-isometric higman embeddings

Abstract: We survey recent results about asymptotic functions of groups, obtained by the authors in collaboration with J.-C. Birget, V. Guba and E. Rips. We also discuss methods used in the proofs of these results.

The chinese monoid

Abstract: Resume: Cet article presente une etude combinatoire du monoide Chinois, un monoide ternaire proche du monoide plaxique, fonde sur le schema cba≡bca≡cab. Un algorithme proche de l'algorithme de Schensted nous permet de caracteriser les classes d'equivalence et d'exhiber une section du monoide. Nous enoncons egalement une correspondance de Robinson–Schensted pour le monoide Chinois avant de nous interesser au calcul du cardinal de certaines classes. Ce travail a permis de developper de nouveaux outils combinatoires. Entre autres, nous avons trouve un plongement de chacune des classes d'equivalence dans la plus grande classe. Quant a la derniere partie de cet article, elle presente l'etude des relations de conjugaison. This paper presents a combinatorial study of the Chinese monoid, a ternary monoid related to the plactic monoid and based on the relation scheme cba≡bca≡cab. An algorithm similar to Schensted's algorithm yields a characterization of the equivalence classes and a cross-section theorem. We also establish a Robinson–Schensted correspondence for the Chinese monoid before computing the order of specific Chinese classes. For this work, we had to develop some new combinatorial tools. Among other things we discovered an embedding of every equivalence class in the largest one. Finally, the end of this paper is devoted to the study of conjugacy classes.

Inevitable graphs and profinite topologies: some solutions to algorithmic problems in monoid and automata theory, stemming from group theory

Abstract: This paper deals with several algorithmic problems in monoid and automata theory arising from group theory. For H a pseudovariety of groups, we give a characterization of the regular elements of the H-kernel of a finite monoid. In particular, we show that if the extension problem for partial one-to-one maps for H is decidable, then so is the set of regular elements of the H-kernel. The extension problem for partial one-to-one maps for H asks if there is an algorithm to determine, given a finite set X and a set S of partial one-to-one maps on X, whether there is a finite set Y containing X so that each of the maps of S can be extended to permutations of Y in such a manner that the group generated by these permutations is in H. This problem is decidable for the pseudovariety of p-groups and nilpotent groups. We explore some other examples here. We also show that if the above problem is decidable, then so is the membership problem for JⓜH. Some applications to the membership problem for J*H are given. Finally, we show that certain pseudovarieties of groups, including the pseudovarieties of p-groups for p prime, are hyperdecidable. The techniques used here lay the groundwork for several future results on problems of this nature.

Conjugation in tree automorphism groups

Abstract: It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

The combination theorem and quasiconvexity

Abstract: We show that if G is a fundamental group of a finite k-acylindrical graph of groups where every vertex group is word-hyperbolic and where every edge-monomorphism is a quasi-isometric embedding, then all the vertex groups are quasiconvex in G (the group G is word-hyperbolic by the Combination Theorem of M. Bestvina and M. Feighn). This allows one, in particular, to approximate the word metric on G by normal forms for this graph of groups.

Normal forms for free aperiodic semigroups

Abstract: The implicit operation ω is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates Using ω there is a well-defined algebra which is known as the free aperiodic semigroup In this article we introduce a specific and rather elementary list of pseudoidentitites, we show that for each n, the n-generated free aperiodic semigroup is defined by this list of pseudoidentities, and then we use this identification to show that it has a decidable word problem In the language of implicit operations, this shows that the pseudovariety of finite aperiodic semigroups is κ-recursive This completes a crucial step towards showing that the Krohn–Rhodes complexity of every finite semigroup is decidable

Intersecting free subgroups in free products of groups

Abstract: A subgroup H of a free product of groups Gα, α∈ I, is termed factor free if for every and β∈I one has SHS-1∩Gβ= {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote , where r(K) is the rank of K. It has earlier been proved by the author that if H, K are finitely generated factor free subgroups of then . It is proved in the article that this estimate is sharp and cannot be improved, that is, there are factor free subgroups H, K in so that and . It is also proved that if the factors Gα, α∈ I, are linearly ordered groups and H, K are finitely generated factor free subgroups of then .

From c-continuations to new quadratic algorithms for automaton synthesis

Abstract: Two classical non-deterministic automata recognize the language denoted by a regular expression: the position automaton which deduces from the position sets defined by Glushkov and McNaughton–Yamada, and the equation automaton which can be computed via Mirkin's prebases or Antimirov's partial derivatives. Let |E| be the size of the expression and ‖E‖ be its alphabetic width, i.e. the number of symbol occurrences. The number of states in the equation automaton is less than or equal to the number of states in the position automaton, which is equal to ‖E‖+1. On the other hand, the worst-case time complexity of Antimirov algorithm is O(‖E‖3· |E|2), while it is only O(‖E‖·|E|) for the most efficient implementations yielding the position automaton (Bruggemann–Klein, Chang and Paige, Champarnaud et al.). We present an O(|E|2) space and time algorithm to compute the equation automaton. It is based on the notion of canonical derivative which makes it possible to efficiently handle sets of word derivatives. By the way, canonical derivatives also lead to a new O(|E|2) space and time algorithm to construct the position automaton.

Simultaneous representations of semilattices by lattices with permutable congruences

Abstract: The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilattice S is isomorphic to the semilattice Conc L of compact congruences of a lattice L. While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that |S|≤ℵ1. Furthermore, one can then take Lwith permutable congruences. This contrasts with the case where |S|≥ℵ2, where there are counterexamples S for which Lcannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well. We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Conc functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square22 can be lifted with respect to the Conc functor. We prove that the latter results cannot be extended to the cube, 23. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Conc functor, by lattices with permutable congruences. We also extend many of our results to lattices with almost permutable congruences, that is, α∨β=αβ ∪βα, for all congruences α and β. We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Conc functor on finite Boolean semilattices.

On growth of Grigorchuk groups

Abstract: We present an analytic technique for estimating the growth for groups of intermediate growth. We apply our technique to Grigorchuk groups, which are the only known examples of such groups. Our estimates generalize and improve various bounds by Grigorchuk, Bartholdi and others.

Profinite semigroups, varieties, expansions and the structure of relatively free profinite semigroups

Abstract: Building on the now generally accepted thesis that profinite semigroups are important to the study of finite semigroups, this paper proposes to apply various of the techniques, already used in studying algebraic semigroups, to profinite semigroups. The goal in mind is to understand free profinite semigroups on a finite set. To do this we define profinite varieties. We then introduce expansions of profinite semigroups, giving examples of several classes of such expansions. These expansions will then be useful in studying various structural properties of relatively free profinite semigroups, since these semigroups will be fixed points of certain expansions. This study also requires a look at profinite categories, semigroupoids, and Cayley graphs, all of which we handle in turn. We also study the structure of the minimal ideal of relatively free profinite semigroups showing, in particular, that the minimal ideal of the free profinite semigroup on a finite set with more than two generators is not a relatively free profinite completely simple semigroup, as well as some generalizations to related pseudovarieties.

Representing braids by automorphisms

Abstract: Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.

Computation in Word-Hyperbolic Groups.

Abstract: We describe two practical algorithms for computing with word-hyperbolic groups, both of which we have implemented. The first is a method for estimating the maximum width, if it exists, of geodesic bigons in the Cayley graph of a finitely presented group G. Our procedure will terminate if and only this maximum width exists, and it has been proved by Papasoglu that this is the case if and only if G is word-hyperbolic. So the algorithm amounts to a method of verifying the property of word-hyperbolicity of G. The aim of the second algorithm is to compute the thinness constant for geodesic triangles in the Cayley graph of G. This seems to be a much more difficult problem, but our implementation does succeed with straightforward examples. Both algorithms involve substantial computations with finite state automata.

Lower bounds on the growth of a group acting on the binary rooted tree

Abstract: In 1980, Rostislav Grigorchuk constructed an infinite finitely generated torsion 2-group G, called the first Grigorchuk group, and in 1983 showed that it is of intermediate growth, with the following estimates on its growth function γ (See [6]): where β= log32(31)≈ 0.991. He conjectured that the lower bound is actually tight. In this paper we improve the lower bound to where α≈0.5157, and thus disproves the conjecture.

Free product, profinite topology and finitely generated subgroups

Abstract: We consider the following property for a group G:(RZn)ifH1,…,Hnare finitely generated subgroups of G then the setH1 H2⋯ Hn= {h1 ⋯ hn| h1∈ H1, …,hn∈ Hn}is closed with respect to the profinite topology of G. It is obvious that finite groups and finitely generated commutative groups have the property (RZn). L. Ribes and P. Zalesskiĭ proved that any free group has (RZn). We show that the property (RZn) is stable under the free product operation. We use techniques developed by B. Herwig and D. Lascar on the one hand, R. Gitik on the other hand.

Geometric equivalence of algebras

Abstract: In this paper, we study the geometric equivalence of algebras in several varieties of algebras. We solve some of the problems formulated in [2], in particular, that of geometric equivalence for real-closed fields and finitely generated commutative groups.

On the hyperdecidability of pseudovarieties of groups

Abstract: Clarifying the relation between Ash's (algebraic-combinatorial) proof and Ribes and Zalesski's (topological) proof of the Rhodes Type II conjecture is an intriguing and interesting question which arose when both proofs appeared in the beginning of the 1990s. Attempting to contribute to this clarification, we observe that two sets, each playing a crucial role in one of the proofs, are in fact equal. The equality of these sets allows us to give an alternative proof of part of the main theorem of Ash's paper where the hyperdecidability of the pseudovariety of all finite groups is established.

On homotopical and homological finiteness conditions for finitely presented monoids

Abstract: An example of a finitely presented monoid is given that does not satisfy the homotopical finiteness condition , although it satisfies both the homological finiteness conditions left and right .

A CHARACTERIZATION OF de MORGAN ALGEBRAS

Abstract: In this note we show that every de Morgan algebra is isomorphic to a two-subset algebra, , where P is a set of pairs (X,Y) of subsets of a set I, (X,Y)⊔ (X′,Y′)=(X∩ X′,Y∪ Y′),(X,Y) ⊓ (X′,Y′)=(X∪ X′,Y∩Y′),~(X,Y)= (Y,X), 0P=(I,∅) and 1P=(∅,I). This characterization generalizes a previous result that applied only to a special type of de Morgan algebras called ternary algebras.

On some semigroups of intermediate growth

Abstract: Asymptotic estimates for growth functions of Belyaev–Sesekin–Trofimov's and Okninski's semigroups are obtained In particular, a question of R Grigorchuk about the existence of matrix semigroups with growth rate between polynomial and , is positively answered

Universal cellular automaton over a hexagonal tiling with 3 states

Abstract: A universal three-state three-neighbor cellular automaton will be constructed. The space selected for this cellular automaton is a hexagonal tiling where the cells are in the vertices and the neighbors are the three nearest cells. We define the local transition rule as well as the basic elements that will aid to build digital circuits and, by the way, prove the universality of this cellular automaton. The local transition rule is defined to be isotropic.

Orthodox transversals of regular semigroups

Abstract: Orthodox transversals were introduced by the first author as a generalization of inverse transversals [Comm. Algebra 27(9) (1999), pp. 4275–4288]. One of our aims in this note is to consider the general case of orthodox transversals. The main results are on the sets I and Λ, two components of regular semigroups with orthodox transversals. We prove that the semibands Ī and generated by I and Λ respectively are bands. Also somewhat interesting generalizations of properties on orthodox semigroups are given; another aim in this note is to give some examples illustrating the situations that the class of regular semigroups with orthodox transversals properly includes the class of regular semigroups with inverse transversals as well as the class of orthodox semigroups.

The type set of a variety is not computable

Abstract: Several problems involving the local structure of finite algebras are shown to be unsolvable by interpreting the halting problem of Turing machines. Specifically, these problems are to decide, given a finite algebra A, whether the type-set of the variety generated by A, which is a subset of {1,2,3,4,5}, omits 2, or 3, or 4, or 5, respectively.

The automorphisms group of the multiplicative cartan decomposition of lie algebra e8

Abstract: We construct a new basis for the exceptional simple Lie algebra L of type E8 and describe the multiplication rule in this basis. It allows to find the action of generators of automorphism group of the multiplicative Cartan decomposition of L on this basis.

ON SMALL CANCELLATION k-GENERATED GROUPS WITH (k-1)-GENERATED SUBGROUPS ALL FREE

Abstract: We give a set of effective sufficient conditions for a group G with small cancellation, generated by k elements, to have (k-1)-generated subgroups all free We also prove that the group G has some more properties that in general do not hold in the whole class of the small cancellation groups

On the freiheitssatz in certain one-relator free products i

Abstract: We study one relator free products in which one of the factors is a cyclic group and the relator has free-product length 4. We find conditions for such presentations to have a Freiheitssatz and classify all non-aspherical presentations under certain conditions.

Torsion-free quotients of braid groups

Abstract: In this paper we show that the braid groups Bn, n≤6, have proper torsion-free non-abelian quotients. We also show that for n≤6 the homotopy braid groups are torsion-free.

The asphericity and freiheitssatz for certain lot-presentations of groups

Abstract: Let U be a word in letters , m >2, and a group G be given by presentation G= . It is proven that this presentation is aspherical provided the word U does not have the form U2 U1, where U1 is a word in letters and U2 is a word in letters . It is also proven that the (images of) x1,…, xm-1 freely generate a free subgroup of G if and only if the word U does not have the foregoing form U2 U1.

On rank, root and equations in free groups

Abstract: Let h1, h2,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w1, w2, …, where each wi is a word in hi and possibly other hj, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that , where is the subgroup generated by the roots of the elements in H. If H0 is finitely generated and the sequence of subgroups H0, H1, H2, … satisfies then the sequence stabilizes, i.e. for some m, Hi=Hi+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" the exponents ai satisfy gcd(a1,…,an)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.