×

Realizable ranks of joins and intersections of subgroups in free groups. (English) Zbl 1452.20019

Author’s abstract: The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups \(H\) and \(K\) of a non-abelian free group. It is an interesting question to “quantify” this bound with respect to the rank of \(H \vee K\), the subgroup generated by \(H\) and \(K\). We describe a set of realizable values \((rk(H \vee K), rk(H \cap K))\) for arbitrary \(H\), \(K\), and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for \(H\) and \(K\) with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of \( H \vee K\), \(H \cap K\) are not realizable, thus resolving the remaining open case \(m = 4\) of Guzman’s “Group-Theoretic Conjecture” in the affirmative. This in turn implies the validity of the corresponding “Geometric Conjecture” on hyperbolic \(3\)-manifolds with a \(6\)-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when \(rk(H) = 2\).

MSC:

20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
20F65 Geometric group theory
57M07 Topological methods in group theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agol, I., Culler, M. and Shalen, P. B., Singular surfaces, mod 2 homology, and hyperbolic volume. I. Trans. Amer. Math. Soc.362(7) (2010) 3463-3498. · Zbl 1195.57037
[2] Bassino, F., Nicaud, C. and Weil, P., Random generation of finitely generated subgroups of a free group, Int. J. Algebra Comput.18(2) (2008) 375-405. · Zbl 1193.05017
[3] G. Baumslag, A. G. Myasnikov and V. Shpilrain, Open problems in combinatorial and geometric group theory, http://www.sci.ccny.cuny.edu/ shpil/gworld/problems/oproblems.html. · Zbl 0976.20023
[4] Bogopolski, O., Introduction to Group Theory, () (European Mathematical Society (EMS), Zürich, 2008), pp. x+177. · Zbl 1215.20001
[5] Culler, M. and Shalen, P. B., 4-free groups and hyperbolic geometry, J. Topol.5(1) (2012) 81-136. · Zbl 1244.57030
[6] Dicks, W., Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture, Invent. Math.117 (1994) 373-389. · Zbl 0809.05055
[7] W. Dicks, Joel Friedman’s Proof of the Strengthened Hanna Neumann Conjecture, Memoirs of the American Mathematical Society, Vol. 233, No. 1100 (American Mathematical Society, 2015), pp. 91-101.
[8] W. Dicks, Simplified Mineyev, http://mat.uab.cat/ dicks/SimplifiedMineyev.pdf.
[9] Diestel, R., Graph Theory, 5th edn, , Vol. 173 (Springer, Berlin, 2017), pp. xviii+428. · Zbl 1375.05002
[10] M. Fayers, The genyoungtabtikz package, version 1.14, http://www.maths.qmul.ac.uk/mf/genyoungtabtikz.html.
[11] Friedman, J., Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture, , Vol. 233, No. 1100 (American Mathematical Society, 2015), pp. xii+106. With an Appendix by Warren Dicks. · Zbl 1327.20025
[12] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.10.0 (2018), https://www.gap-system.org.
[13] Guzman, R. K., Hyperbolic 3-manifolds with \(k\)-free fundamental group, Topology Appl.173 (2014) 142-156. · Zbl 1302.57053
[14] Guzman, R. K. and Shalen, P. B., The geometry of \(k\)-free hyperbolic 3-manifolds, J. Topol. Anal., https://doi.org/10.1142/S1793525320500016 · Zbl 1469.57021
[15] J. E. Hunt, The Hanna Neumann conjecture and the rank of the join, https://arxiv.org/abs/1509.04449.
[16] Imrich, W. and Müller, T., On Howson’s theorem. Arch. Math. (Basel)62(3) (1994) 193-198. · Zbl 0812.20009
[17] Ivanov, S. V., On a conjecture of Imrich and Müller, J. Group Theory20(4) (2017) 823-828. · Zbl 1388.20042
[18] Ivanov, S. V., On joins and intersections of subgroups in free groups, J. Comb. Algebra2 (2018) 1-18. · Zbl 1491.20067
[19] Jaikin-Zapirain, A., Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Math. J.166 (2017) 1955-1987. · Zbl 1375.20035
[20] Kapovich, I. and Myasnikov, A., Stallings foldings and subgroups of free groups, J. Algebra248(2) (2002) 608-668. · Zbl 1001.20015
[21] Kent, R. P. IV, Achievable ranks of intersections of finitely generated free groups, Int. J. Algebra Comput.15(2) (2005) 339-341. · Zbl 1072.20034
[22] Kent, R. P. IV, Intersections and joins of free groups, Algebr. Geom. Topol.9(1) (2009) 305-325. · Zbl 1170.20017
[23] Louder, L. and McReynolds, D. B., Graphs of subgroups of free groups, Algebr. Geom. Topol.9(1) (2009) 327-355. · Zbl 1185.20026
[24] Mineyev, I., Groups, graphs, and the Hanna Neumann conjecture, J. Topol. Anal.4(1) (2012) 1-12. · Zbl 1257.20034
[25] C. Sievers, FGA (Free Group Algorithms) — A GAP package, Version 1.4.0 (2018), http://www.icm.tu-bs.de/ag_algebra/software/FGA.
[26] Stallings, J. R., Topology of finite graphs. Invent. Math.71(3) (1983) 551-565. · Zbl 0521.20013
[27] T. Tantau, The TikZ and PGF Packages, Manual for Version 3.1, http://sourceforge.net/projects/pgf, 2019-01-05.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.