×

zbMATH — the first resource for mathematics

Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. (English) Zbl 0809.05055
Let \(G\) be a free group. The reduced rank of \(G\) is defined as \(\widetilde r(G)= \max\{\text{rank}(G)- 1,0\}.\) W. D. Neumann [Lect. Notes Math. 1456, 161-170 (1990; Zbl 0722.20016)] asked if the following statement is true: If \(H\) and \(K\) are subgroups of \(G\) then \[ \sum_{HgK\in H\backslash G/K}\widetilde r(H^ g\cap K)\leq \widetilde r(H) \widetilde r(K). \] This statement is called the strengthened Hanna Neumann conjecture because it is a stronger version of a still unsolved conjecture made by Hanna Neumann [Publ. Math. 4, 186-189 (1956; Zbl 0070.02001)]. Here \(H^ g\) denotes \(g^{-1} Hg\), and \(H\backslash G/K\) denotes the set of double cosets \(HgK\), \(g\in G\).
In this article it is shown that this new conjecture is equivalent to the following previously uncontemplated conjecture, called the amalgamated graph conjecture: If \(\Delta\) is any finite bipartite graph for which there exist three finite bipartite \(\Delta\)-graphs \(A\), \(B\), \(\Gamma\) such that the finite bipartite graph \((A\vee_ \Delta B)\vee (B\vee_ \Delta \Gamma)\vee (\Gamma\vee_ \Delta A)\) is simple-edged and is the disjoint union of two isomorphic bipartite subgraphs, then \(\Delta\) has at most half as many edges as the complete bipartite graph with the same vertex set. Terminology used here is as follows: a bipartite graph has each vertex coloured black or white, and each edge joins two vertices of different colours; a \(\Delta\)-graph is a graph given with a specified copy of \(\Delta\) as a bipartite subgraph; simple-edged means two vertices are joined by at most one edge, \(\vee\) denotes disjoint union, and \(\vee_ \Delta\) the pushout identifying the two copies of \(\Delta\). Simple proofs of some previously known results are given to demonstrate the strength of this equivalence.

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E05 Free nonabelian groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Burns, R.G.: On the intersection of finitely generated subgroups of a free group. Math. Z.119, 121-130 (1971) · Zbl 0204.34003 · doi:10.1007/BF01109964
[2] Dicks, W., Dunwoody, M.J.: Groups acting on graphs (Camb. Stud. Adv. Math., vol. 17) Cambridge: Cambridge University Press 1989 · Zbl 0665.20001
[3] Gersten, S.M.: Intersections of finitely generated subgroups of free groups and resolutions of graphs. Invent. Math.71 567-591 (1983) · Zbl 0521.20014 · doi:10.1007/BF02095994
[4] Howson, A.G.: On the intersection of finitely generated free groups. J. Lond. Math. Soc.29, 428-434 (1954) · Zbl 0056.02106 · doi:10.1112/jlms/s1-29.4.428
[5] Imrich, W.: On finitely generated subgroups of free groups. Arch. Math.28, 21-24 (1977) · Zbl 0385.20016 · doi:10.1007/BF01223883
[6] Imrich, W.: Subgroup theorems and graphs. In: Little, C.H.C. (ed.) Combinatorial Mathematics V, Melbourne 1976. (Lect. Notes Math., vol. 622, pp. 1-27), Berlin Heidelberg New York: Springer 1977
[7] Neumann, H.: On intersections of finitely generated subgroups of free groups. Publ. Math., Debrecen4, 186-189 (1956) · Zbl 0070.02001
[8] Neumann, H.: On intersections of finitely generated subgroups of free groups. Addendum. Publ. Math., Debrecen5, 128 (1958) · Zbl 0085.25304
[9] Neumann, W.D.: On intersections of finitely generated subgroups of free groups. In: Kovács, L.G. (ed.) Groups?Canberra 1989. (Lect. Notes Math., vol. 1456, pp. 161-170) Berlin Heidelberg New York: Springer 1990
[10] Nickolas, P.: Intersections of finitely generated free groups. Bull. Aust. Math. Soc.31, 339-348 (1985) · Zbl 0579.20018 · doi:10.1017/S0004972700009291
[11] Reidemeister, K.: Einführing in die kombinatorische Topologie. Braunschweig 1932; reprinted: New York: Chelsea 1950 · JFM 58.0611.01
[12] Serre, J.P.: Arbres, amalgames, SL2. (Astérisque, no. 46) Paris: Soc. Math. Fr. 1977; English translation: Trees. Berlin Heidelberg New York: Springer 1980
[13] Servatius, B.: A short proof of a theorem of Burns. Math. Z.184, 133-137 (1983) · Zbl 0509.20014 · doi:10.1007/BF01162012
[14] Stallings, J.R.: On torsion-free groups with infinitely many ends. Ann. Math.88, 312-334 (1968) · Zbl 0238.20036 · doi:10.2307/1970577
[15] Stallings, J.R.: Topology of finite graphs. Invent. Math.71, 551-565 (1983) · Zbl 0521.20013 · doi:10.1007/BF02095993
[16] Tardos, G.: On the intersection of subgroups of a free group. Invent. Math.108, 29-36 (1992) · Zbl 0798.20015 · doi:10.1007/BF02100597
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.