2007/02/13 by Warren Dicks, WARREN DICKS, S. V. Ivanov +1
Mathematics · #Advanced Operator Algebra Research #Finite Group Theory Research #Geometric and Algebraic Topology #math.GR #msc:20E06
paper · pdf · doi:10.1017/s0305004107001041
published as Math. Proc. Cambridge Philos. Soc. 144(2008), 511-534. · 28 pages, no figures
arxiv created 2007/02/13 · openalex publication_date 2008/05/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/04
Abstract Let ( G i | i ∈ I ) be a family of groups, let F be a free group, and let G = F ∗ \mathop\Large *i∈ I Gi, the free product of F and all the G i . Let 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 ∩ Gig = \1\. By the Kurosh Subgroup Theorem, every element of 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\ ∈ \naturals ∪ \∞\ ⊆ [0,∞]. To avoid the vacuous case, we make the additional assumption that 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 ∈ [\fun, 2\fun] , and, moreover, \upp = 2\fun whenever G has 2-torsion. Since \upp is finite, F is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that \upp = \fun whenever G does not have 2-torsion.