Illinois Journal of Mathematics

On the intersection of free subgroups in free products of groups with no $2$-torsion

Warren Dicks and S. V. Ivanov

Full-text: Open access

Abstract

Let $(G_\ell\mid\ell\in L)$ be a family of groups and let $F$ be a free group. Let $G$ denote $F \ast {\Large{*}}_{\ell\in L} G_\ell$, the free product of $F$ and all the $G_\ell$. Let $\mathcal{F}$ denote the set of all finitely generated (free) subgroups $H$ of $G$ which have the property that, for each $g \in G$ and each $\ell\in L$, $H \cap G_\ell^{g} = \{1\}$. For each free group $H$, the reduced rank of $H$ is defined as $\bar{r}(H) := \max\{\operatorname{rank}(H) -1, 0\} \in[0,\infty]$. Set \begin{eqnarray*} \theta & := & \max\biggl\{ \biggl\vert\frac{\lvert D \rvert}{\lvert D \rvert-2}\biggr\vert : \\ &&{}\mbox{$D$ is a finite subgroup of $G$ with }\lvert D \rvert \ne2 \biggr\} \in [1,3],\\ \sigma & := & \inf\biggl\{ s \in[0,\infty] : \mbox{for all } H, K \in\mathcal{F}, \\ &&{}\sum_{HgK \in H \setminus G /K}\bar{r} (H^g\cap K) \le s {\theta} \bar{r} (H) \bar{r} (K) \biggr\} \in [0,\infty]. \end{eqnarray*} We are interested in precise bounds for $\sigma$. If every element of $\mathcal{F}$ is cyclic, then $\sigma = 0$. We henceforth assume that some element of $\mathcal{F}$ has rank two.

In the case where $G=F$ and, hence, $v = 1$, Hanna Neumann and Walter Neumann proved that $\sigma\in[1,2]$ and it is a famous conjecture that $\sigma = 1$, called the Strengthened Hanna Neumann Conjecture.

For the general case, we proved that $\sigma \in[1,2]$ and if $G$ has 2-torsion then $\sigma = 2$. We conjectured that if $G$ is 2-torsion-free then $\sigma = 1$. In this article, we prove the following implications which show that under certain circumstances $\sigma \lt 2$.

If $G$ is $2$-torsion-free and has $3$-torsion, then $\sigma \le \frac{8}{7}$.

If $G$ is $2$-torsion-free and $3$-torsion-free and has $5$-torsion, then $\sigma \le \frac{9}{5}$.

If $p$ is an odd prime number and $G = C_p \ast C_p$, then $\sigma \le 2 - \frac{(4+2\sqrt{3})p}{(2 p -3 + \sqrt{3})^2}$. In particular, if $G = C_3 \ast C_3$ then $\sigma = 1$, and if $G = C_5 \ast C_5$ then $\sigma \le 1.52$.

Article information

Source
Illinois J. Math., Volume 54, Number 1 (2010), 223-248.

Dates
First available in Project Euclid: 9 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1299679747

Digital Object Identifier
doi:10.1215/ijm/1299679747

Mathematical Reviews number (MathSciNet)
MR2776994

Zentralblatt MATH identifier
1226.20020

Subjects
Primary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations
Secondary: 20F06: Cancellation theory; application of van Kampen diagrams [See also 57M05] 20F32

Citation

Dicks, Warren; Ivanov, S. V. On the intersection of free subgroups in free products of groups with no $2$-torsion. Illinois J. Math. 54 (2010), no. 1, 223--248. doi:10.1215/ijm/1299679747. https://projecteuclid.org/euclid.ijm/1299679747


Export citation

References

  • W. Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Stud. Adv. Math., vol. 17, Cambridge University Press, Cambridge, 1989. Errata at http://mat.uab.cat/~dicks/DDerr.html.
  • W. Dicks and S. V. Ivanov, On the intersection of free subgroups in free products of groups, Math. Proc. Cambridge Philos. Soc. 144 (2008), 511–534.
  • D. J. Grynkiewicz, On extending Pollard's Theorem for $t$-representable sums, Israel J. Math. 177 (2010), 413–439.
  • S. V. Ivanov, On the intersection of finitely generated subgroups in free products of groups, Internat. J. Algebra Comput. 9 (1999), 521–528.
  • S. V. Ivanov, Intersecting free subgroups in free products of groups, Internat. J. Algebra Comput. 11 (2001), 281–290.
  • W. Lück, $L\sp2$-invariants: Theory and applications to geometry and $K$-theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002.
  • H. Neumann, On intersections of finitely generated subgroups of free groups, Publ. Math. Debrecen 4 (1956), 186–189.
  • H. Neumann, On intersections of finitely generated subgroups of free groups. Addendum, Publ. Math. Debrecen 5 (1957), 128.
  • W. D. Neumann, On intersections of finitely generated subgroups of free groups, Groups–-Canberra 1989 (L. G. Kovács, ed.), Lecture Notes Math., vol. 1456, Springer-Verlag, Berlin, 1990, pp. 161–170.
  • J. M. Pollard, A generalisation of the theorem of Cauchy and Davenport, J. London Math. Soc. 8 (1974), 460–462.
  • M. Sykiotis, On subgroups of finite complexity in groups acting on trees, J. Pure Appl. Algebra 200 (2005), 1–23.
  • G. Tardos, On the intersection of subgroups of a free group, Invent. Math. 108 (1992), 29–36.