On the rank of intersection of subgroups of a free product of groups
type: Item
instance of: scholarly article
title: On the rank of intersection of subgroups of a free product of groups
zbMATH Open document ID: 0896.20021
DOI: 10.1016/S0022-4049(98)00120-0
author: J. Ning
published in: Journal of Pure and Applied Algebra
publication date: 1998-10-06T00:00:00Z
review text: The authors study the intersection of two subgroups of a free product of two groups. It is well-known that the intersection of two finitely generated subgroups of a free group is again finitely generated and there is a bound of its rank in terms of the ranks of these subgroups, given by \textit{A. G. Howson} in 1954 [J. Lond. Math. Soc. 29, 428-434 (1954; Zbl 0056.02106)]. This was improved in 1958 by H. Neumann. This property is called the Howson property. In case of a free product of groups one has the Kurosh rank \(\text{Krk }H\) of a subgroup \(H\), as the number of free factors arising from the Kurosh subgroup theorem of free products. So the authors extend the above-mentioned result of free groups in case of subgroups of a free product of two groups \(A\) and \(B\). Their main result states that: If \(H,K\leq A*B\) have finite Kurosh ranks, then \[ \text{KrK}(H\cap K)-1\leq 2(\text{Krk }H-1)(\text{Krk }K-1)+2\min\{\text{Krk }H-1,\text{Krk }K-1\}. \] This improves a similar result of \textit{T. Soma} [J. Pure Appl. Algebra 66, No. 1, 81-95 (1990; Zbl 0713.20020)]. The proof is geometric. The authors first restate the Kurosh subgroup theorem for the free product \(A*B\) and then define the ``Kurosh graph'' of a subgroup \(H\) of \(A*B\) arising actually from the realization of \(H\) as the fundamental group of a certain surface complex.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1121645
zbMATH Keywords: rank of subgroups
zbMATH Keywords: intersections of subgroups
zbMATH Keywords: Howson property
zbMATH Keywords: free products of groups
zbMATH Keywords: Kurosh rank
zbMATH Keywords: numbers of free factors
zbMATH Keywords: Kurosh subgroup theorem
MaRDI profile type: Publication
cites work: Intersections of Finitely Generated Subgroups in Free Products
cites work: Q3342738
cites work: On the Intersection of Finitely Generated Free Groups
cites work: Q3243302
cites work: Q3209315
cites work: Intersection of finitely generated surface groups
Wikidata QID: Q126556773
full work available at URL: s0022-4049(98)00120-0
OpenAlex ID: W2092222018
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description: scientific article; zbMATH DE number 1121645