scholarly journals GENERICITY OF FILLING ELEMENTS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250008 ◽  
Author(s):  
BRENT B. SOLIE
Keyword(s):  
Free Group ◽  
Time Complexity ◽  
Linear Time ◽  
Sufficient Condition ◽  
Membership Problem ◽  
Isometric Action ◽  
Finitely Generated ◽  
Generic Subset ◽  
Translation Length

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F(X) on an ℝ-tree. We give a proof that the set of filling elements of F(X) is exponentially F(X)-generic in the sense of Arzhantseva and Ol'shanskiı. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F(X)-generic subset consisting only of filling elements and whose membership problem has linear time complexity.

scholarly journals A FAST ALGORITHM FOR STALLINGS' FOLDING PROCESS

2006 ◽  
Vol 16 (06) ◽  
pp. 1031-1045 ◽  
Author(s):  
NICHOLAS W. M. TOUIKAN
Keyword(s):  
Free Group ◽  
Fast Algorithm ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Folding Process ◽  
Algorithmic Problems ◽  
The Given

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J1,…,Jm〉 of a finitely generated nonabelian free group F = F(x1,…,xn) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log *(N)), where N is the sum of the word lengths of the given generators of H.

scholarly journals Almost congruence extension property for subgroups of free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 125-146
Author(s):  
Lev Glebsky ◽  
Nevarez Nieto Saul
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Free Groups ◽  
Extension Property ◽  
Normal Closure ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Finitely Generated ◽  
Finite Set

AbstractLetHbe a subgroup ofFand{\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}}the normal closure ofHinF. We say thatHhas the Almost Congruence Extension Property (ACEP) inFif there is a finite set of nontrivial elements{\digamma\subset H}such that for any normal subgroupNofHone has{H\cap\langle\kern-1.422638pt\langle N\rangle\kern-1.422638pt\rangle_{F}=N}whenever{N\cap\digamma=\emptyset}. In this paper, we provide a sufficient condition for a subgroup of a free group to not possess ACEP. It also shows that any finitely generated subgroup of a free group satisfies some generalization of ACEP.

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

1998 ◽  
Vol 08 (02) ◽  
pp. 235-294 ◽  
Author(s):  
Jean-Camille Birget
Keyword(s):  
Word Problem ◽  
Time Complexity ◽  
Linear Time ◽  
Algorithm Design ◽  
Embedding Theorem ◽  
Algebraic Characterization ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Isoperimetric Function ◽  
Finitely Presented

The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem: Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) ≥T(n)+T(m)). Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is O (T(∣x∣+∣y∣)2). Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(To)) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(To)). In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function ≤ D (where D(n) ≥ n), then the word problem of S has nondeterministic time complexity O(D). The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T) O(1) )) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) O(1) ) isoperimetric function. An algorithmic problem L is in NP (or more generally, in NTIME((T) O(1) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) O(1) ) isoperimetric function. In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

CLOSED SUBGROUPS IN PRO-V TOPOLOGIES AND THE EXTENSION PROBLEM FOR INVERSE AUTOMATA

2001 ◽  
Vol 11 (04) ◽  
pp. 405-445 ◽  
Author(s):  
S. MARGOLIS ◽  
M. SAPIR ◽  
P. WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Finite Groups ◽  
Extension Problem ◽  
Membership Problem ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Finite Monoid ◽  
Finite Set ◽  
One To One

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. Résumé: Nous établissons un lien entre le problème du calcul de l'adhéerence d'un sous-groupe finiment engendré du groupe libre dans la topologie pro-V, oú V est une pseudovariété de groupes finis, et un probléme d'extension pour les automates inversifs qui peut être énoncé de la faç con suivante: étant données des transformations partielles injectives d'un ensemble fini, peuvent-elles être étendues en des permutations qui engendrent un groupe dans V? Les deux problèmes sont équivalents si V est fermée par extensions. Nous intéressant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adhérence pro-p d'un sous-groupe finiment engendré du groupe libre en temps polynomial et pour calculer effectivement sa clôture pro-nilpotente. Enfin nous appliquons nos résultats à un problème de théorie des monoïdes finis, celui de de l'appartenance dans les pseudovariétés de monoïdes inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariété de groupes.

ALGORITHMIC PROBLEMS ON INVERSE MONOIDS OVER VIRTUALLY FREE GROUPS

2008 ◽  
Vol 18 (01) ◽  
pp. 181-208 ◽  
Author(s):  
VOLKER DIEKERT ◽  
NICOLE ONDRUSCH ◽  
MARKUS LOHREY
Keyword(s):  
Word Problem ◽  
Linear Time ◽  
Free Groups ◽  
Hyperbolic Group ◽  
Inverse Monoid ◽  
Membership Problem ◽  
Finitely Generated ◽  
Arbitrary Group ◽  
First Order Theory ◽  
Rational Subset

Let G be a finitely generated virtually-free group. We consider the Birget–Rhodes expansion of G, which yields an inverse monoid and which is denoted by IM (G) in the following. We show that for a finite idempotent presentation P, the word problem of a quotient monoid IM (G)/P can be solved in linear time on a RAM. The uniform word problem, where G and the presentation P are also part of the input, is EXPTIME-complete. With IM (G)/P we associate a relational structure, which contains for every rational subset L of IM (G)/P a binary relation, consisting of all pairs (x,y) such that y can be obtained from x by right multiplication with an element from L. We prove that the first-order theory of this structure is decidable. This result implies that the emptiness problem for Boolean combinations of rational subsets of IM (G)/P is decidable, which, in turn implies the decidability of the submonoid membership problem of IM (G)/P. These results were known previously for free groups, only. Moreover, we provide a new algorithmic approach for these problems, which seems to be of independent interest even for free groups. We also show that one cannot expect decidability results in much larger frameworks than virtually-free groups because the subgroup membership problem of a subgroup H in an arbitrary group G can be reduced to a word problem of some IM (G)/P, where P depends only on H. A consequence is that there is a hyperbolic group G and a finite idempotent presentation P such that the word problem is undecidable for some finitely generated submonoid of IM (G)/P. In particular, the word problem of IM (G)/P is undecidable.

scholarly journals FULLY RESIDUALLY FREE GROUPS AND GRAPHS LABELED BY INFINITE WORDS

2006 ◽  
Vol 16 (04) ◽  
pp. 689-737 ◽  
Author(s):  
ALEXEI G. MYASNIKOV ◽  
VLADIMIR N. REMESLENNIKOV ◽  
DENIS E. SERBIN
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Limit Groups ◽  
Infinite Words ◽  
Integer Coefficients ◽  
Ring Of Polynomials

Let F = F(X) be a free group with basis X and ℤ[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (ℤ[t],X)-graphs — a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich–Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon's group Fℤ[t], the author's representation of elements of Fℤ[t] by infinite (ℤ[t],X)-words, and Stallings folding method for subgroups of free groups. As an application, we solve the membership problem for finitely generated subgroups of Fℤ[t], as well as for finitely generated fully residually free groups.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

scholarly journals A NOTE ON THE HOWSON PROPERTY IN INVERSE SEMIGROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 457-463 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

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