On the rank of the intersection of free subgroups in virtually free groups.
type: Item
instance of: scholarly article
title: On the rank of the intersection of free subgroups in virtually free groups.
zbMATH Open document ID: 1307.20024
published in: Journal of Algebra
publication date: 2014-09-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.3115
reviewed by: Dimitrios Varsos
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 6341134
zbMATH Keywords: virtually free groups
zbMATH Keywords: amalgamated free products
zbMATH Keywords: HNN-extensions
zbMATH Keywords: graphs of groups
zbMATH Keywords: Bass-Serre theory
zbMATH Keywords: Hanna Neumann inequality
zbMATH Keywords: reduced ranks
zbMATH Keywords: intersections of finitely generated subgroups
zbMATH Keywords: Hanna Neumann conjecture
MaRDI profile type: Publication
OpenAlex ID: W1984788336
arXiv ID: 1301.3115
cites work: On Cayley graphs of virtually free groups.
cites work: Introduction to group theory. Translated from the Russian. With a new chapter.
cites work: Q3344126
cites work: On the intersection of free subgroups in free products of groups
cites work: CONTEXT-FREE GROUPS AND THEIR STRUCTURE TREES
cites work: Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks
cites work: On the Intersection of Finitely Generated Free Groups
cites work: ON THE INTERSECTION OF FINITELY GENERATED SUBGROUPS IN FREE PRODUCTS OF GROUPS
cites work: On the Kurosh rank of the intersection of subgroups in free products of groups.
cites work: Finite and infinite cyclic extensions of free groups
cites work: When Is a Graph Product of Groups Virtually-Free?
cites work: Groups, graphs, and the Hanna Neumann conjecture.
cites work: Q3243302
cites work: Q3341041
cites work: An estimate for the rank of the intersection of subgroups in free amalgamated products of two groups with normal finite amalgamated subgroup.
DOI: 10.1016/J.JALGEBRA.2014.07.010
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author: Alexander Zakharov
review text: Let \(F\) be a free group and \(H,K\) finitely generated subgroups of \(F\). In [\textit{J. Friedman}, Mem. Am. Math. Soc. 1100 (2015; Zbl 1327.20025)] and [\textit{I. Mineyev}, J. Topol. Anal. 4, No. 1, 1-12 (2012; Zbl 1257.20034)] it is proved Hanna Neumann's conjecture which states that NEWLINE\[NEWLINE\overline r(H\cap K)\leq\overline r(H)\overline r(K);NEWLINE\]NEWLINE where \(\overline r(H)=\max (r(H)-1,0)\) is the reduced rank of a free group \(H\) of (free) rank \(r(H)\).NEWLINENEWLINE In [\textit{S. V. Ivanov}, Int. J. Algebra Comput. 9, No. 5, 521-528 (1999; Zbl 1028.20021)] a relevant result is proved for free products. Namely if \(G=G_1*G_2\) is the free product of the groups \(G_1\) and \(G_2\), \(H\) and \(K\) are finitely generated subgroups of \(G\) which intersect trivially with all the conjugates to the factors \(G_1\) and \(G_2\) (therefore are free), then the following estimate holds: NEWLINE\[NEWLINE\overline r(H\cap K)\leq 6\overline r(H)\overline r(K).NEWLINE\]NEWLINE In [Sb. Math. 204, No. 2, 223-236 (2013); translation from Mat. Sb. 204, No. 2, 73-86 (2013; Zbl 1285.20027)] the author generalized the result above and proved an analogous result for the rank of the intersection of free subgroups in free products with amalgamation of a finite normal subgroup.NEWLINENEWLINE Here the author extents the previous results in a larger class of groups. More precisely he proves the Theorem. Suppose \(G\) is the fundamental group of a finite graph of groups \((\Gamma,Y)\) with finite edge groups \(G_e\). Let \(H,K\leq G\) be finitely generated subgroups which intersect trivially with the conjugates to all the vertex groups of \((\Gamma,Y)\) (and are, therefore, free). Then the following estimate holds: NEWLINE\[NEWLINE\overline r(H\cap K)\leq 6m\cdot\overline r(H)\overline r(K),\tag{*}NEWLINE\]NEWLINE where NEWLINE\[NEWLINEm=\max\{|g^{-1}G_eg\cap HK|,\;e\in E(Y),\;g\in G\}.NEWLINE\]NEWLINE In particular, NEWLINE\[NEWLINE\overline r(H\cap K)\leq 6m'\cdot…
description: scientific article; zbMATH DE number 6341134