Linear programing and the intersection of free subgroups in free products of groups

2017 ◽  
Vol 11 (4) ◽  
pp. 1113-1177
Author(s):  
Sergei Ivanov
Keyword(s):  
Free Products ◽  
Products Of Groups ◽  
Free Subgroups

scholarly journals On the intersection of free subgroups in free products of groups

2008 ◽  
Vol 144 (3) ◽  
pp. 511-534 ◽  
Author(s):  
WARREN DICKS ◽  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Additional Assumption ◽  
Free Products ◽  
Reduced Rank ◽  
Products Of Groups ◽  
Image Position ◽  
Free Subgroups ◽  
Special Case ◽  
Subgroup Theorem

AbstractLet (Gi | i ∈ I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi.Let $\mathcal{F}$ denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ ∈ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we define Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

scholarly journals INTERSECTING FREE SUBGROUPS IN FREE PRODUCTS OF GROUPS

2001 ◽  
Vol 11 (03) ◽  
pp. 281-290 ◽  
Author(s):  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Product Formula ◽  
Finitely Generated ◽  
Free Products ◽  
Ordered Groups ◽  
Products Of Groups ◽  
Free Subgroups

A subgroup H of a free product [Formula: see text] of groups Gα, α∈ I, is termed factor free if for every [Formula: see text] and β∈I one has SHS-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 has earlier been proved by the author that if H, K are finitely generated factor free subgroups of [Formula: see text] then [Formula: see text]. It is proved in the article that this estimate is sharp and cannot be improved, that is, there are factor free subgroups H, K in [Formula: see text] so that [Formula: see text] and [Formula: see text]. It is also proved that if the factors Gα, α∈ I, are linearly ordered groups and H, K are finitely generated factor free subgroups of [Formula: see text] then [Formula: see text].

scholarly journals Intersecting free subgroups in free products of left ordered groups

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Sergei V. Ivanov
Keyword(s):  
Free Products ◽  
Ordered Groups ◽  
Products Of Groups ◽  
Free Subgroups

AbstractA conjecture of Dicks and the author on the rank of the intersection of factor-free subgroups in free products of groups is proved for the case of left ordered groups.

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.

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
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