Remarks on the Intersection of Finitely Generated Subgroups of a Free Group

1986 ◽  
Vol 29 (2) ◽  
pp. 204-207 ◽  
Author(s):  
R. G. Burns ◽  
Wilfried Imrich ◽  
Brigitte Servatius
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Short Proof ◽  
Finitely Generated ◽  
Modest Improvement ◽  
General Bound

AbstractThe first result gives a (modest) improvement of the best general bound known to date for the rank of the intersection U ∩ V of two finite-rank subgroups of a free group F in terms of the ranks of U and V. In the second result it is deduced from that bound that if A is a finite-rank subgroup of F and B < F is non-cyclic, then the index of A ∩ B in B, if finite, is less than 2(rank(A) - 1), whence in particular if rank (A) = 2, then B ≤ A. (This strengthens a lemma of Gersten.) Finally a short proof is given of Stallings' result that if U, V (as above) are such that U ∩ V has finite index in both U and V, then it has finite index in their join 〈U, V〉.

scholarly journals RANDOM GENERATION OF FINITELY GENERATED SUBGROUPS OF A FREE GROUP

2008 ◽  
Vol 18 (02) ◽  
pp. 375-405 ◽  
Author(s):  
FRÉDÉRIQUE BASSINO ◽  
CYRIL NICAUD ◽  
PASCAL WEIL
Keyword(s):  
Uniform Distribution ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Efficient Algorithm ◽  
Random Generation ◽  
Finitely Generated ◽  
Average Case ◽  
Case Complexity ◽  
Average Case Complexity

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity [Formula: see text] in the RAM model and [Formula: see text] in the bitcost model.

Groups in which every Finitely Generated Subgroup is almost a Free Factor

1979 ◽  
Vol 31 (6) ◽  
pp. 1329-1338 ◽  
Author(s):  
A. M. Brunner ◽  
R. G. Burns
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Products

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

scholarly journals A note on groups with separable finitely generated subgroups

1987 ◽  
Vol 36 (1) ◽  
pp. 153-160 ◽  
Author(s):  
R. G. Burns ◽  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Cyclic Extension ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Positive Direction ◽  
One Relator Groups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

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.

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

scholarly journals Subgroups of free solvable groups

1971 ◽  
Vol 12 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Jacques Lewin ◽  
Tekla Lewin
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Finite Index ◽  
Solvable Groups ◽  
Free Subgroup ◽  
Infinite Index ◽  
Finitely Generated ◽  
Derived Length ◽  
Inner Automorphisms ◽  
Torsion Free

A consequence of Schreier's formula is that if G is a subgroup of the free group F of rank n < 1 and rank G ≦ n, then G = F or G is of infinite index in F. However, if S is a free sovlvable group of derived length I < 1 and H is a subgroup of S which is free solvable of the same length, then the rank of H does not exceed the rank of S. These observations led G. Baumslag to conjecture that if H is of finite index in S then H = S. In fact, we have sharper results in two directions. If H and S are free solvable of the same length, not only is H of infinite index in S, but δ1−1(S)/δ1−1(H) is torsion-free. In another direction we need not assume that S is free solvable, only that s is torsion-free and of derived length l (l > 1) and that H is not cyclic. Thus Stallings' theorem [11] that a finitely generated torsionfree group with a free subgroup of finite index is itself free has an even stronger counterpart in the variety of groups solvable of length at most l (l > 1): a torsionfree group in that variety with a non-cyclic free subgroup of finite index coincides with this subgroup. The proof relies on the following theorem: If S is a free solvable group, J is the group of automorphisms of S which induce the identity on S/S', and I is the group of inner automorphisms of S, then J/I is torsion-free. The proofs of these theorems form the bulk of the first four sections.

scholarly journals On the Residual Finiteness of Some Generalized Products of Soluble Groups of Finite Rank

2015 ◽  
Vol 20 (1) ◽  
pp. 124-132
Author(s):  
A. V. Rozov
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Finite Rank ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Residually Finite ◽  
Amalgamated Subgroup

Let G be a free product of residually finite virtually soluble groups A and B of finite rank with an amalgamated subgroup H, H 6= A and H 6= B. And let H contains a subgroup W of finite index which is normal in both A and B. We prove that the group G is residually finite if and only if the subgroup H is finitely separable in A and B. Also we prove that if all subgroups of A and B are finitely separable in A and B, respectively, all finitely generated subgroups of G are finitely separable in G.

Finitely generated subgroups of free groups as formal languages and their cogrowth

2021 ◽  
Vol volume 13, issue 2 ◽  
Author(s):  
Arman Darbinyan ◽  
Rostislav Grigorchuk ◽  
Asif Shaikh
Keyword(s):  
Free Group ◽  
Finite Automaton ◽  
Finite Rank ◽  
Regular Language ◽  
Free Groups ◽  
Formal Languages ◽  
Finitely Generated ◽  
Deterministic Finite Automaton ◽  
Method Of Calculation ◽  
Reduced Words

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

scholarly journals Folding-like techniques for CAT(0) cube complexes

2021 ◽  
pp. 1-12
Author(s):  
MICHAEL BEN–ZVI ◽  
ROBERT KROPHOLLER ◽  
RYLEE ALANZA LYMAN
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Coxeter Groups ◽  
Free Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Seminal Paper ◽  
Finite Graphs ◽  
Cube Complexes ◽  
Positively Curved

Abstract In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

Sign in / Sign up

Export Citation Format

Share Document