scholarly journals On finitely generated submonoids of virtually free groups

2018 ◽  
Vol 10 (2) ◽  
pp. 63-82
Author(s):  
Pedro V. Silva ◽  
Alexander Zakharov
Keyword(s):  
Word Problem ◽  
Free Group ◽  
Free Groups ◽  
Isomorphism Problem ◽  
Geometric Characterization ◽  
Finitely Generated ◽  
Free Monoids ◽  
Virtually Free Group

AbstractWe prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.

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.

REFINING KNOWN RESULTS ON THE GENERALIZED WORD PROBLEM FOR FREE GROUPS

1992 ◽  
Vol 02 (02) ◽  
pp. 221-236 ◽  
Author(s):  
IAIN A. STEWART
Keyword(s):  
Word Problem ◽  
Free Groups ◽  
Complexity Classes ◽  
Finitely Generated ◽  
Complete Problem ◽  
Complete Problems

We refine the known result that the generalized word problem for finitely-generated subgroups of free groups is complete for P via logspace reductions and show that by restricting the lengths of the words in any instance and by stipulating that all words must be conjugates then we obtain complete problems for the complexity classes NSYMLOG, NL, and P. The proofs of our results range greatly: some are complexity-theoretic in nature (for example, proving completeness by reducing from another known complete problem), some are combinatorial, and one involves the characterization of complexity classes as problems describable in some logic.

scholarly journals QUASI-ISOMETRY AND FINITE PRESENTATIONS OF LEFT CANCELLATIVE MONOIDS

2013 ◽  
Vol 23 (05) ◽  
pp. 1099-1114 ◽  
Author(s):  
ROBERT D. GRAY ◽  
MARK KAMBITES
Keyword(s):  
Word Problem ◽  
Main Tool ◽  
Geometric Characterization ◽  
Finitely Generated ◽  
Solvable Word Problem ◽  
Finite Presentability ◽  
Finite Presentations ◽  
Solvable Word ◽  
Cancellative Monoids

We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.

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 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 Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

scholarly journals ON RESTRICTING SUBSETS OF BASES IN RELATIVELY FREE GROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250030
Author(s):  
LUCAS SABALKA ◽  
DMYTRO SAVCHUK
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Relatively Free Group

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.

MALNORMALITY IS DECIDABLE IN FREE GROUPS

1999 ◽  
Vol 09 (06) ◽  
pp. 687-692 ◽  
Author(s):  
GILBERT BAUMSLAG ◽  
ALEXEI MYASNIKOV ◽  
VLADIMIR REMESLENNIKOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated

We prove here that there is an algorithm whereby one can decide whether or not any finitely generated subgroup of a finitely generated free group is malnormal.

scholarly journals The isomorphism problem for finitely generated fully residually free groups

2007 ◽  
Vol 208 (3) ◽  
pp. 961-977 ◽  
Author(s):  
Inna Bumagin ◽  
Olga Kharlampovich ◽  
Alexei Miasnikov
Keyword(s):  
Free Groups ◽  
Isomorphism Problem ◽  
Finitely Generated

scholarly journals Amenable minimal Cantor systems of free groups arising from diagonal actions

2017 ◽  
Vol 2017 (722) ◽  
Author(s):  
Yuhei Suzuki
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Crossed Products ◽  
Virtually Free Group

AbstractWe study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined

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