scholarly journals Orders on Trees and Free Products of Left-ordered Groups

2020 ◽  
Vol 63 (2) ◽  
pp. 335-347
Author(s):  
Warren Dicks ◽  
Zoran Šunić
Keyword(s):  
Free Product ◽  
Short Proof ◽  
Free Products ◽  
Ordered Groups ◽  
Vertex Set ◽  
Total Orders

AbstractWe construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.As an application, we provide a short proof (modulo Bass–Serre theory) of Vinogradov’s result that the free product of left-orderable groups is left-orderable.

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 Right orders and amalgamation for lattice-ordered groups

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
V. Bludov ◽  
A. Glass
Keyword(s):  
Free Product ◽  
Sufficient Conditions ◽  
State University ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Amalgamated Subgroup ◽  
Necessary And Sufficient

AbstractLet H i be a sublattice subgroup of a lattice-ordered group G i (i = 1, 2). Suppose that H 1 and H 2 are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G 1 and G 2 can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H 1 and H 1φ. In this article we prove that the group free product of G 1 and G 2 amalgamating H 1 and H 1φ is right orderable and so embeddable (as a group) in a lattice-orderable group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable [BLUDOV, V. V.—GLASS, A. M. W.: Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. (3) 99 (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal (ℓ-ideals) and solve of Problem 1.42 from [KOPYTOV, V. M.—MEDVEDEV, N. YA.: Ordered groups. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].

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].

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

scholarly journals Free product decompositions in images of certain free products of groups

2007 ◽  
Vol 310 (1) ◽  
pp. 57-69
Author(s):  
N.S. Romanovskii ◽  
John S. Wilson
Keyword(s):  
Free Product ◽  
Free Products ◽  
Products Of Groups

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.

Residual finiteness of free products of combinatorial strict inverse semigroups

1994 ◽  
Vol 124 (1) ◽  
pp. 137-147 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Inverse Semigroups ◽  
Residual Finiteness ◽  
Free Products ◽  
Residually Finite

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

scholarly journals A five lemma for free products of groups with amalgamations

1970 ◽  
Vol 3 (1) ◽  
pp. 85-96 ◽  
Author(s):  
J. L. Dyer
Keyword(s):  
Free Product ◽  
Free Products ◽  
Products Of Groups ◽  
Five Lemma

This paper explores a five-lemma situation in the context of a free product of a family of groups with amalgamated subgroups (that is, a colimit of an appropriate diagram in the category of groups). In particular, for two families {Aα}, {Bα} of groups with amalgamated subgroups {Aαβ}, {Bαβ} and free products A, B we assume the existence of homomorphisms Aα → Bα whose restrictions Aαβ → Bαβ are isomorphisms and which induce an isomorphism A → B between the products. We show that the usual five-lemma conclusion is false, in that the morphisms Aα → Bα are in general neither monic nor epic. However, if all Bα → B are monic, Aα → Bα is always epic; and if Aα → A is monic, for all α, then Aα → Bα is an isomorphism.

scholarly journals Free products of topological groups with a closed subgroup amalgamated

1986 ◽  
Vol 40 (3) ◽  
pp. 414-420 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Closed Subgroup ◽  
Countable Family ◽  
Topological Groups ◽  
Free Products ◽  
Dense Image ◽  
Amalgamated Free Product

AbstractIt is shown that if {Gn: n = 1, 2,…} is a countable family of Hausdorff kω-topological groups with a common closed subgroup A, then the topological amalgamated free product *AGn exists and is a Hausdorff kω-topological group with each Gn as a closed subgroup. A consequence is the theorem of La Martin that epimorphisms in the category of kω-topological groups have dense image.

A remark about the description of free products of groups

1966 ◽  
Vol 62 (2) ◽  
pp. 129-134 ◽  
Author(s):  
John Stallengs
Keyword(s):  
Free Product ◽  
Binary Operation ◽  
Free Products ◽  
Products Of Groups ◽  
Reduced Words

The free product A* B of groups A and B can be described in two ways.We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .

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