scholarly journals Schreier systems in free products

1965 ◽  
Vol 7 (2) ◽  
pp. 61-79 ◽  
Author(s):  
I. M. S. Dey
Keyword(s):  
Free Group ◽  
Free Products ◽  
Cancellation Property

In 1927 Schreier [8] proved the Nielsen-Schreier Theorem that a subgroup H of a free group F is a free group by selecting a left transversal for H in F possessing a certain cancellation property. Hall and Rado [5] call a subset T of a free group F a Schreier system in F if it possesses this cancellation property, and consider the existence of a subgroup H of F such that a given Schreier system T is a left transversal for H in F.

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 AUTOMATA OVER A BINARY ALPHABET GENERATING FREE GROUPS OF EVEN RANK

2011 ◽  
Vol 21 (01n02) ◽  
pp. 329-354 ◽  
Author(s):  
BENJAMIN STEINBERG ◽  
MARIYA VOROBETS ◽  
YAROSLAV VOROBETS
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Finite Automata ◽  
Free Groups ◽  
Free Products ◽  
Cyclic Groups ◽  
Binary Alphabet

We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

scholarly journals The minimally displaced set of an irreducible automorphism is locally finite

2020 ◽  
Vol 55 (2) ◽  
pp. 301-336
Author(s):  
Stefano Francaviglia ◽  
◽  
Armando Martino ◽  
Dionysios Syrigos ◽  
◽  
...  
Keyword(s):  
Free Group ◽  
Outer Space ◽  
General Setting ◽  
Deformation Space ◽  
Train Track ◽  
Free Products ◽  
Deformation Spaces ◽  
Locally Finite ◽  
Irreducible Automorphism

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

scholarly journals On the intersection of free subgroups in free products of groups

2008 ◽  
Vol 144 (3) ◽  
pp. 511-534 ◽  
Author(s):  
WARREN DICKS ◽  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Additional Assumption ◽  
Free Products ◽  
Reduced Rank ◽  
Products Of Groups ◽  
Image Position ◽  
Free Subgroups ◽  
Special Case ◽  
Subgroup Theorem

AbstractLet (Gi | i ∈ I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi.Let $\mathcal{F}$ denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ ∈ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we define Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

scholarly journals INTERSECTIONS OF SUBGROUPS IN VIRTUALLY FREE GROUPS AND VIRTUALLY FREE PRODUCTS

2019 ◽  
Vol 101 (2) ◽  
pp. 266-271
Author(s):  
ANTON A. KLYACHKO ◽  
ANASTASIA N. PONFILENKO
Keyword(s):  
Free Product ◽  
Free Group ◽  
Short Proof ◽  
Free Groups ◽  
Free Subgroup ◽  
Free Products ◽  
Virtually Free Group ◽  
Free Subgroups

This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.

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 ON THE LOCAL-INDICABILITY COHEN–LYNDON THEOREM

2011 ◽  
Vol 53 (3) ◽  
pp. 637-656 ◽  
Author(s):  
YAGO ANTOLÍN ◽  
WARREN DICKS ◽  
PETER A. LINNELL
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Induction Method ◽  
Free Products ◽  
Detailed Review ◽  
Generating Set ◽  
Inductive Proof ◽  
Trivial Element ◽  
One Relator Groups ◽  
Indicable Group

AbstractFor a group H and a subset X of H, we let HX denote the set {hxh−1 | h ∈ H, x ∈ X}, and when X is a free-generating set of H, we say that the set HX is a Whitehead subset of H. For a group F and an element r of F, we say that r is Cohen–Lyndon aspherical in F if F{r} is a Whitehead subset of the subgroup of F that is generated by F{r}. In 1963, Cohen and Lyndon (D. E. Cohen and R. C. Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526–537) independently showed that in each free group each non-trivial element is Cohen–Lyndon aspherical. Their proof used the celebrated induction method devised by Magnus in 1930 to study one-relator groups. In 1987, Edjvet and Howie (M. Edjvet and J. Howie, A Cohen–Lyndon theorem for free products of locally indicable groups, J. Pure Appl. Algebra45 (1987), 41–44) showed that if A and B are locally indicable groups, then each cyclically reduced element of A*B that does not lie in A ∪ B is Cohen–Lyndon aspherical in A*B. Their proof used the original Cohen–Lyndon theorem. Using Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem, one can deduce the local-indicability Cohen–Lyndon theorem: if F is a locally indicable group and T is an F-tree with trivial edge stabilisers, then each element of F that fixes no vertex of T is Cohen–Lyndon aspherical in F. Conversely, by Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem are immediate consequences of the local-indicability Cohen–Lyndon theorem. In this paper we give a detailed review of a Bass–Serre theoretical form of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen–Lyndon theorem that uses neither Magnus induction nor the original Cohen–Lyndon theorem. We conclude with a review of some standard applications of Cohen–Lyndon asphericity.

scholarly journals INTERSECTING FREE SUBGROUPS IN FREE PRODUCTS OF GROUPS

2001 ◽  
Vol 11 (03) ◽  
pp. 281-290 ◽  
Author(s):  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Product Formula ◽  
Finitely Generated ◽  
Free Products ◽  
Ordered Groups ◽  
Products Of Groups ◽  
Free Subgroups

A subgroup H of a free product [Formula: see text] of groups Gα, α∈ I, is termed factor free if for every [Formula: see text] 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 [Formula: see text], 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 [Formula: see text] then [Formula: see text]. 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 [Formula: see text] so that [Formula: see text] and [Formula: see text]. 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 [Formula: see text] then [Formula: see text].

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