Abstract length functions in groups

1976 ◽  
Vol 80 (3) ◽  
pp. 451-463 ◽  
Author(s):  
I. M. Chiswell
Keyword(s):  
Free Product ◽  
Free Group ◽  
Free Groups ◽  
Arbitrary Group ◽  
Free Products ◽  
Length Functions ◽  
Normal Word ◽  
Free Product Of Groups ◽  
Fixed Basis ◽  
Product Of Groups

If F is a free group on some fixed basis X, there is a mapping from F to the non-negative integers, given by sending an element of F to the length of the normal word in X±1 representing it. A similar mapping is obtained in the case of a free product of groups. Lyndon (3) considered mappings from an arbitrary group to the non-negative integers having certain properties in common with these mappings on free groups and free products.

scholarly journals The palindromic width of a free product of groups

2006 ◽  
Vol 81 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Valery Bardakov ◽  
Vladimir Tolstykh
Keyword(s):  
Free Product ◽  
Free Groups ◽  
Uniform Bound ◽  
Free Products ◽  
Cyclic Groups ◽  
Similar Fact ◽  
Products Of Groups ◽  
Free Product Of Groups ◽  
Verbal Subgroups ◽  
Product Of Groups

AbstractPalindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

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 Faithful, irreducible *-representations for group algebras of free products

1999 ◽  
Vol 42 (3) ◽  
pp. 559-574 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Irreducible Representation ◽  
Free Product ◽  
Irreducible Representations ◽  
Group Algebras ◽  
Free Products ◽  
Characteristic 2 ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups A and B, where |A|≥3 and |B|≥2. We construct faithful, irreducible *-representations for the group algebras ℂ[G] and ℓ1(G). The construction gives a faithful, irreducible representation for F[G] when the field F does not have characteristic 2.

Cyclic Subgroup Separability of Generalized Free Products

1993 ◽  
Vol 36 (3) ◽  
pp. 296-302 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Free Product ◽  
Cyclic Subgroup ◽  
Free Products ◽  
Cyclic Subgroups ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Generalized Free Products ◽  
Product Of Groups

AbstractWe derive a criterion for a generalized free product of groups to be cyclic subgroup separable. We see that most of the known results for cyclic subgroup separability are covered by this criterion, and we apply the criterion to polygonal products of groups. We show that a polygonal product of finitely generated abelian groups, amalgamating cyclic subgroups, is cyclic subgroup separable.

ON THE INTERSECTION OF FINITELY GENERATED SUBGROUPS IN FREE PRODUCTS OF GROUPS

1999 ◽  
Vol 09 (05) ◽  
pp. 521-528 ◽  
Author(s):  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Free Groups ◽  
Product Formula ◽  
Finitely Generated ◽  
Free Products ◽  
Products Of Groups ◽  
Free Subgroups

A subgroup H of a free product [Formula: see text] of groups Gα, α∈ I, is called factor free if for every [Formula: see text] and β ∈ I one has S H S-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 is proven that if H, K are finitely generated factor free subgroups of a free product [Formula: see text] then [Formula: see text]. It is also shown that the inequality [Formula: see text] of Hanna Neumann conjecture on subgroups of free groups does not hold for factor free subgroups of free products.

scholarly journals On subgroups of amalgamated free products

1971 ◽  
Vol 69 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Edward T. Ordman
Keyword(s):  
Free Product ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Free Product Of Groups ◽  
Product Of Groups

In 1934 Kurosh(9) proved that ‘a subgroup of a free product of groups is again a free product’. Many other proofs of this, and attempts to generalize it to amalgamated free products, have appeared (e.g. (7), (1), (10) and (8)). Recently the theory of groupoids has been applied to this area with increasing success. In 1966 Higgins (6) used groupoids to prove the generalization of Grushko's Theorem (3) due to Wagner (14).

scholarly journals Free products of inverse semigroups II

1991 ◽  
Vol 33 (3) ◽  
pp. 373-387 ◽  
Author(s):  
Peter R. Jones ◽  
Stuart W. Margolis ◽  
John Meakin ◽  
Joseph B. Stephen
Keyword(s):  
Free Product ◽  
General Result ◽  
Commutative Diagram ◽  
Free Semigroup ◽  
Inverse Semigroups ◽  
Canonical Forms ◽  
Free Products ◽  
Free Product Of Groups ◽  
Product Of Groups

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.

All automorphisms of free groups with maximal rank fixed subgroups

1996 ◽  
Vol 119 (4) ◽  
pp. 615-630 ◽  
Author(s):  
D. J. Collins ◽  
E. C. Turner
Keyword(s):  
Growth Rate ◽  
Free Product ◽  
Free Group ◽  
Linear Growth ◽  
Free Groups ◽  
Maximal Rank ◽  
Free Products ◽  
Automorphisms Of Free Groups ◽  
Free Rank ◽  
Fixed Subgroups

The Scott Conjecture, proven by Bestvina and Handel [2] says that an automorphism of a free group of rank n has fixed subgroup of rank at most n. We characterise in Theorem A below those automorphisms that realise this maximum. It follows from this characterisation, for example, that any such automorphism has linear growth. In our paper [3], we generalised the Scott Conjecture to arbitrary free products, using Kuros rank (see Section 2 below) in place of free rank; in Theorem B, we characterise those automorphisms of a free product realising the maximum. We show that in this case the growth rate is also linear. These results extend those of [4].

Free products and residual nilpotency

1969 ◽  
Vol 1 (1) ◽  
pp. 11-13 ◽  
Author(s):  
I. M. S. Dey
Keyword(s):  
Free Product ◽  
Lower Central Series ◽  
Central Series ◽  
Free Products ◽  
Residual Nilpotency ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups Gα, [Gα] the cartesian subgroup of G and k [Gα] the intersection of [G] with the k–th term of the lower central series for G. Then the k[Gα] form a descending chain of subgroups of ] and it is shown that if the intersection of all the subgroups in this chain is trivial then G and hence each Gα, is residually nilpotent. This answers a question of S. Moran.

scholarly journals Relative hyperbolicity for automorphisms of free products and free groups

2020 ◽  
pp. 1-38
Author(s):  
François Dahmani ◽  
Ruoyu Li
Keyword(s):  
Free Product ◽  
Free Group ◽  
Semidirect Product ◽  
Abelian Groups ◽  
Free Groups ◽  
Conjugacy Problem ◽  
Product Formula ◽  
Free Products ◽  
Relatively Hyperbolic ◽  
Relative Hyperbolicity

We prove that for a free product [Formula: see text] with free factor system [Formula: see text], any automorphism [Formula: see text] preserving [Formula: see text], atoroidal (in a sense relative to [Formula: see text]) and none of whose power send two different conjugates of subgroups in [Formula: see text] on conjugates of themselves by the same element, gives rise to a semidirect product [Formula: see text] that is relatively hyperbolic with respect to suspensions of groups in [Formula: see text]. We recover a theorem of Gautero–Lustig and Ghosh that, if [Formula: see text] is a free group, [Formula: see text] an automorphism of [Formula: see text], and [Formula: see text] is its family of polynomially growing subgroups, then the semidirect product by [Formula: see text] is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.

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