TY - JOUR

T1 - On the intersection of free subgroups in free products of groups

AU - Dicks, Warren

AU - Ivanov, S. V.

N1 - Funding Information:
Acknowledgements. The research of the first-named author was jointly funded by the MEC (Spain) and the EFRD (EU) through Projects BFM2003-06613 and MTM2006-13544. The research of the second-named author was partially funded by the NSF (USA) through grants DMS 00-99612 and DMS 04-00746.

PY - 2008/5

Y1 - 2008/5

N2 - Let (Gi | i I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi. Let $\mathcal{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, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define $ \begin{eqnarray} \upp:= \sup \bigg\{\frac{\barr(H\cap K)} {\barr (H) \cdot\barr(K)} : H, K \in \mathcal{F} \text{ and} \barr (H) \cdot \barr(K) \neq 0\bigg\} \in [1,\infty]. \end{eqnarray}$ 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 $ \fun:= \max\bigg\{\frac{\abss{L}}{ \abss{L}-2} : \text{$L$ is a subgroup of $G$ and } \abss{L} \ne 2 \bigg\} \in[1,3].$ Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

AB - Let (Gi | i I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi. Let $\mathcal{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, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define $ \begin{eqnarray} \upp:= \sup \bigg\{\frac{\barr(H\cap K)} {\barr (H) \cdot\barr(K)} : H, K \in \mathcal{F} \text{ and} \barr (H) \cdot \barr(K) \neq 0\bigg\} \in [1,\infty]. \end{eqnarray}$ 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 $ \fun:= \max\bigg\{\frac{\abss{L}}{ \abss{L}-2} : \text{$L$ is a subgroup of $G$ and } \abss{L} \ne 2 \bigg\} \in[1,3].$ Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

UR - http://www.scopus.com/inward/record.url?scp=43449124755&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43449124755&partnerID=8YFLogxK

U2 - 10.1017/S0305004107001041

DO - 10.1017/S0305004107001041

M3 - Article

AN - SCOPUS:43449124755

VL - 144

SP - 511

EP - 534

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -