scholarly journals Aut-invariant quasimorphisms on free products

2021 ◽  
Author(s):  
Bastien Karlhofer
Keyword(s):  
Free Product ◽  
Dihedral Group ◽  
Free Products ◽  
Infinite Dimensional ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractLet $$G=A *B$$ G = A ∗ B be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these 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.

Group algebras of free products of finite groups which are CS-rings

2018 ◽  
pp. 1850224
Author(s):  
Shoichi Kondo
Keyword(s):  
Free Product ◽  
Group Algebra ◽  
Finite Groups ◽  
Dihedral Group ◽  
Group Algebras ◽  
Free Products ◽  
Cyclic Groups ◽  
Algebra Over A Field ◽  
Cs Rings

This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field [Formula: see text] is a CS-ring in case the orders of two groups are not zero in [Formula: see text]. Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition [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.

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 Surface groups, infinite generating sets, and stable commutator length

2019 ◽  
Vol 150 (5) ◽  
pp. 2379-2386
Author(s):  
Dan Margalit ◽  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Surface Group ◽  
Mapping Class ◽  
Surface Groups ◽  
Closed Curves ◽  
Generating Set ◽  
Generating Sets ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractWe give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

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.

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