scholarly journals Commutator length of powers in free products of groups

2021 ◽  
pp. 1-18
Author(s):  
Vadim Yu. Bereznyuk ◽  
Anton A. Klyachko
Keyword(s):  
Free Products ◽  
Products Of Groups ◽  
Commutator Length

Abstract Given groups $A$ and $B$ , what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$ ). Other similar problems are also considered.

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

scholarly journals Quasiperiodic and mixed commutator factorizations in free products of groups

2018 ◽  
Vol 50 (5) ◽  
pp. 832-844 ◽  
Author(s):  
Sergei V. Ivanov ◽  
Anton A. Klyachko
Keyword(s):  
Free Products ◽  
Products Of Groups

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.

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.

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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