scholarly journals A free-group valued invariant of free knots

2021 ◽  
pp. 2150042
Author(s):  
Vassily Olegovich Manturov
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Free Products ◽  
Cyclic Groups ◽  
Virtual Knots

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups).

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.

Test Elements, Generic Elements and Almost Primitivity in Free Products

2016 ◽  
Vol 23 (02) ◽  
pp. 263-280
Author(s):  
Ann-Kristin Engel ◽  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Free Groups ◽  
Free Products ◽  
Cyclic Groups ◽  
Test Elements ◽  
Generic Elements

In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free products and in particular to free products of cyclic groups.

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 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.

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.

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.

On Almost Primitive Elements of Free Groups With an Application to Fuchsian Groups

1993 ◽  
Vol 45 (2) ◽  
pp. 225-254 ◽  
Author(s):  
A. M. Brunner ◽  
R. G. Burns ◽  
Sheila Oates-Williams
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Proper Subgroup ◽  
Fuchsian Groups ◽  
Infinite Index ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Free Products ◽  
Cyclic Groups ◽  
Do So

AbstractAn element of a free group F is called almost primitive in F, if it is primitive in every proper subgroup containing it, though not in F itself. Several examples of almost primitive elements (APEs) are exhibited. The main results concern the behaviour of proper powers wℓ of certain APEs w in a free group F (and, more generally, in free products of cycles) with respect to any subgroup H containing such a power “minimally“: these assert, in essence, that either such powers of w behave in H as do powers of primitives of F, or, if not, then they “almost” do so and furthermore H must then have finite index in F precisely determined by the smallest positive powers of conjugates of w lying in H. Finally, these results are applied to show that the groups of a certain class (potentially larger than that of finitely generated Fuchsian groups) have the property that all their subgroups of infinité index are free products of cyclic groups.

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].

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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