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.

INEVITABLE GRAPHS AND PROFINITE TOPOLOGIES: SOME SOLUTIONS TO ALGORITHMIC PROBLEMS IN MONOID AND AUTOMATA THEORY, STEMMING FROM GROUP THEORY

2001 ◽  
Vol 11 (01) ◽  
pp. 25-71 ◽  
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Group Theory ◽  
Nilpotent Groups ◽  
Automata Theory ◽  
Extension Problem ◽  
Membership Problem ◽  
Regular Elements ◽  
Finite Set ◽  
Algorithmic Problems ◽  
One To One

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.

scholarly journals STALLINGS FOLDINGS AND SUBGROUPS OF AMALGAMS OF FINITE GROUPS

2007 ◽  
Vol 17 (08) ◽  
pp. 1493-1535 ◽  
Author(s):  
L. MARKUS-EPSTEIN
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finite Automaton ◽  
Finite Automata ◽  
Free Groups ◽  
Minimal Immersion ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Folding Algorithm ◽  
Algorithmic Problems

Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

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 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 ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1611-1634 ◽  
Author(s):  
ABDÓ ROIG ◽  
ENRIC VENTURA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Finite Rank ◽  
Polynomial Algorithm ◽  
Minimum Size ◽  
Input Word ◽  
Finitely Generated ◽  
Factor Problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved in polynomial time.

scholarly journals COMPRESSED WORDS AND AUTOMORPHISMS IN FULLY RESIDUALLY FREE GROUPS

2010 ◽  
Vol 20 (03) ◽  
pp. 343-355 ◽  
Author(s):  
JEREMY MACDONALD
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Free Group ◽  
Polynomial Time ◽  
Free Groups ◽  
Finitely Generated

We show that the compressed word problem in a finitely generated fully residually free group ([Formula: see text]-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an [Formula: see text]-group is decidable in polynomial time.

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.

scholarly journals Finite and infinite cyclic extensions of free groups

1973 ◽  
Vol 16 (4) ◽  
pp. 458-466 ◽  
Author(s):  
A. Karrass ◽  
A. Pietrowski ◽  
D. Solitar
Keyword(s):  
Finite Number ◽  
Free Group ◽  
Finite Groups ◽  
Free Groups ◽  
Finite Extension ◽  
Finitely Generated ◽  
Free Products ◽  
Tree Product ◽  
Image Position ◽  
Generalized Free Products

Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

scholarly journals Partial Automata and Finitely Generated Congruences: An Extension of Nerode's Theorem

1992 ◽  
Vol 21 (400) ◽  
Author(s):  
Dexter Kozen
Keyword(s):  
Polynomial Time ◽  
Term Rewriting ◽  
Equivalent System ◽  
Finitely Generated ◽  
Ground Term ◽  
One To One ◽  
Rewrite System

Let T_Sigma be the set of ground terms over a finite ranked alphabet Sigma. We define <em> partial autornata on</em> T_Sigma and prove that the finitely generated congruences on T_Sigma are in one-to one correspondence (up to isomorphism) with the finite partial automata on Sigma with no inaccessible and no inessential states. We give an application in term rewriting: every ground term rewrite system has a canonical equivalent system that can be constructed in polynomial time.

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