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 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 Groups whose three-generator subgroups are free

1989 ◽  
Vol 40 (2) ◽  
pp. 163-174 ◽  
Author(s):  
Gilbert Baumslag ◽  
Peter B. Shalen
Keyword(s):  
Free Groups ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Products Of Groups

We define a certain class of groups, Ck, which we show to contain the class of all k-free groups. Our main theorem shows that certain amalgamated free products of groups in C3, are again in C3. In the appendix we show that many 3-manifold groups belong to Ck for suitable k.

Second Order Dehn Functions for Amalgamated Free Products of Groups

2009 ◽  
Vol 16 (04) ◽  
pp. 699-708
Author(s):  
Xiaofeng Wang ◽  
Xiaomin Bao
Keyword(s):  
Free Product ◽  
Upper Bound ◽  
Second Order ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Amalgamated Free Product ◽  
Finite Set

A finite set of generators for a free product of two groups of type F3with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

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.

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