On the intersection of free subgroups in free products of groups
| dc.creator | Dicks, Warren | |
| dc.creator | Ivanov, S. V. | |
| dc.date | 2007-02-13 | |
| dc.date.accessioned | 2026-07-07T09:38:40Z | |
| dc.date.available | 2026-07-07T09:38:40Z | |
| dc.description | Let (G_i | i in I) be a family of groups, let F be a free group, and let G = F *(*I G_i), the free product of F and all the G_i. Let FF denote the set of all finitely generated subgroups H of G which have the property that, for each g in G and each i in I, H \cap G_i^{g} = {1}. By the Kurosh Subgroup Theorem, every element of FF is a free group. For each free group H, the reduced rank of H is defined as r(H) = max{rank(H) -1, 0} in \naturals \cup {\infty} \subseteq [0,\infty]. To avoid the vacuous case, we make the additional assumption that FF contains a non-cyclic group, and we define sigma := sup{r(H\cap K)/(r(H)r(K)) : H, K in FF and r(H)r(K) \ne 0}, sigma in [1,\infty]. We are interested in precise bounds for sigma. In the special case where I is empty, Hanna Neumann proved that sigma in [1,2], and conjectured that sigma = 1; almost fifty years later, this interval has not been reduced. With the understanding that \infty/(\infty -2) = 1, we define theta := max{|L|/(|L|-2) : L is a subgroup of G and |L| > 2}, theta in [1,3]. Generalizing Hanna Neumann's theorem, we prove that sigma in [theta, 2 theta], and, moreover, sigma = 2 theta if G has 2-torsion. Since sigma is finite, FF is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that sigma = theta whenever G does not have 2-torsion. | |
| dc.description | 28 pages, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0702363 | |
| dc.identifier | http://arxiv.org/abs/math/0702363 | |
| dc.identifier | Math. Proc. Cambridge Philos. Soc. 144(2008), 511-534. | |
| dc.identifier | doi:10.1017/S0305004107001041 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160887 | |
| dc.subject | Group Theory | |
| dc.subject | 20E06 | |
| dc.title | On the intersection of free subgroups in free products of groups | |
| dc.type | text |