Generation of amalgamated free products of cyclic groups by finite automata over minimal alphabet

2020 ◽  
Author(s):  
Veronika Prokhorchuk
Keyword(s):  
Finite Automata ◽  
Free Products ◽  
Cyclic Groups ◽  
Amalgamated Free Products

scholarly journals Stable commutator length in amalgamated free products

2015 ◽  
Vol 07 (04) ◽  
pp. 693-717 ◽  
Author(s):  
Tim Susse
Keyword(s):  
Geometric Model ◽  
Free Products ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Torus Knot ◽  
Knot Complements ◽  
Stable Commutator Length ◽  
Free Abelian Groups ◽  
Commutator Length ◽  
Boundary Mapping

We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.

Units in Group Rings of Free Products of Prime Cyclic Groups

1998 ◽  
Vol 50 (2) ◽  
pp. 312-322 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Maria Lucia Sobral Singer
Keyword(s):  
Unit Group ◽  
Group Rings ◽  
Rational Group ◽  
Free Products ◽  
Structural Result ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Finite Subgroups ◽  
Zassenhaus Conjecture

AbstractLet G be a free product of cyclic groups of prime order. The structure of the unit group U(ℚG) of the rational group ring ℚG is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of U(ℚG), up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in ℤ G is proved. A strong version of the Tits Alternative for U(ℚG) is obtained as a corollary of the structural result.

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 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 ON THE GROWTH OF GROUPS AND AUTOMORPHISMS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 869-874 ◽  
Author(s):  
MARTIN R. BRIDSON
Keyword(s):  
Finitely Generated ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Growth Functions ◽  
Growth Of Groups

We consider the growth functions βΓ(n) of amalgamated free products Γ = A *C B, where A ≅ B are finitely generated, C is free abelian and |A/C| = |A/B| = 2. For every d ∈ ℕ there exist examples with βΓ(n) ≃ nd+1βA(n). There also exist examples with βΓ(n) ≃ en. Similar behavior is exhibited among Dehn functions.

scholarly journals Separating conjugates in amalgamated free products and HNN extensions

1980 ◽  
Vol 29 (1) ◽  
pp. 35-51 ◽  
Author(s):  
Joan L. Dyer
Keyword(s):  
Nilpotent Groups ◽  
Conjugacy Classes ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Nontrivial Center ◽  
One Relator Groups ◽  
Amalgamated Subgroup ◽  
Conjugacy Separable ◽  
Base Group

AbstractA group G is termed conjugacy separable (c.s.) if any pair of distinct conjugacy classes may be mapped to distinct conjugacy classes in some finite epimorph of G. The free product of A and B with cyclic amalgamated subgroup H is shown to be c.s. if A and B are both free, or are both finitely generated nilpotent groups. Further, one-relator groups with nontrivial center and HNN extensions with c.s. base group and finite associated subgroups are also c.s.

On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras

2016 ◽  
Vol 50 (1) ◽  
pp. 39-47
Author(s):  
Qihui Li ◽  
Don Hadwin ◽  
Jiankui Li ◽  
Xiujuan Ma ◽  
Junhao Shen
Keyword(s):  
Free Products ◽  
C Algebras ◽  
Amalgamated Free Products

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

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