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On the intersection of free subgroups in free products of groups

Published online by Cambridge University Press:  01 May 2008

WARREN DICKS  and
S. V. IVANOV
WARREN DICKS
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain. e-mail: dicks@mat.uab.cat
S. V. IVANOV
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana IL 61801, U.S.A. e-mail: ivanov@math.uiuc.edu
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Abstract

Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let the free product of F and all the Gi.

Let denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, By the Kurosh Subgroup Theorem, every element of is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as To avoid the vacuous case, we make the additional assumption that contains a non-cyclic group, and we define We are interested in precise bounds for . In the special case where I is empty, Hanna Neumann proved that ∈ [1,2], and conjectured that = 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 , and, moreover, whenever G has 2-torsion. Since is finite, is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that whenever G does not have 2-torsion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 511 - 534
Copyright
Copyright © Cambridge Philosophical Society 2008

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