×

zbMATH — the first resource for mathematics

Stallings foldings and subgroups of free groups. (English) Zbl 1001.20015
In this long paper, the authors study the subgroups of a free group by using the approach of J. R. Stallings, who introduced the notion of foldings of graphs [in Arboreal group theory, Publ., Math. Sci. Res. Inst. 19, 355-368 (1991; Zbl 0782.20018)]. In their own words “they re-cast in a more combinatorial and computational form the topological approach of J. Stallings to the study of subgroups of free groups”. For this, they give a detailed, selfcontained, elementary and comprehensive treatment of the used approach. They also include “complete and independent proofs of most basic facts, a substantial number of explicit examples and a wide assortment of possible applications”. In doing so they reprove many classical well-known, or folklore results about the subgroup structure of free groups. For example they prove the Takahasi-Higman theorem on ascending chains of subgroups of a free group. More precisely: Let \(F=F(X)\) be a free group of finite rank. Let \(M\geq 1\) be an integer. Then every strictly ascending chain of subgroups of \(F(X)\) of rank at most \(M\) terminates. Concluding we can say that this paper is a good account of the subgroup structure of free groups in this setting. The reference list contains 48 items.

MSC:
20E05 Free nonabelian groups
57M07 Topological methods in group theory
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Baumslag, G.; Myasnikov, A.; Remeslennikov, V., Algebraic geometry over groups. I. algebraic sets and ideal theory, J. algebra, 219, 16-79, (1999) · Zbl 0938.20020
[2] Baumslag, G., Malnormality is decidable in free groups, Internat. J. algebra comput., 9, 687-692, (1999) · Zbl 0949.20024
[3] Bestvina, M.; Feighn, M., Bounding the complexity of simplicial group actions on trees, Invent. math., 103, 449-469, (1991) · Zbl 0724.20019
[4] Bestvina, M.; Feighn, M., A combination theorem for negatively curved groups, J. differential geom., 35, 85-101, (1992) · Zbl 0724.57029
[5] Bestvina, M.; Feighn, M., Addendum and correction to A combination theorem for negatively curved groups, J. differential geom., 43, 783-788, (1996) · Zbl 0862.57027
[6] Bestvina, M.; Handel, M., Train tracks and automorphisms of free groups, Ann. of math. (2), 135, 1-51, (1992) · Zbl 0757.57004
[7] Bestvina, M.; Feighn, M., Train-tracks for surface homeomorphisms, Topology, 34, 109-140, (1995) · Zbl 0837.57010
[8] Bleiler, S.A.; Jones, A.C., The free product of groups with amalgamated subgroup malnormal in a single factor, J. pure appl. algebra, 127, 119-136, (1998) · Zbl 0931.20022
[9] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), Am. Elsevier New York · Zbl 1134.05001
[10] Dicks, W.; Formanek, E., The rank three case of the hanna Neumann conjecture, J. group theory, 4, 113-151, (2001) · Zbl 0976.20011
[11] Dicks, W.; Ventura, E., The group fixed by a family of injective endomorphisms of a free group, (1996), Am. Math. Soc Providence · Zbl 0845.20018
[12] Dicks, W., Equivalence of the strengthened hanna Neumann conjecture and the amalgamated graph conjecture, Invent. math., 117, 373-389, (1994) · Zbl 0809.05055
[13] Dunwoody, M.J., Folding sequences, the Epstein birthday schrift, Geom. topol. (Coventry), 139-158, (1998) · Zbl 0927.20013
[14] Dunwoody, M.J., A small unstable action on a tree, Math. res. lett., 6, 697-710, (1999) · Zbl 1045.20018
[15] Gersten, S.M., Intersections of finitely generated subgroups of free groups and resolutions of graphs, Invent. math., 71, 567-591, (1983) · Zbl 0521.20014
[16] Gersten, S.M., The double exponential theorem for isodiametric and isoperimetric functions, Internat J. algebra comput., 1, 321-327, (1991) · Zbl 0728.20029
[17] Gitik, R., On the combination theorem for negatively curved groups, Internat J. algebra comput., 6, 751-760, (1996) · Zbl 0879.20014
[18] Goldstein, R.Z.; Turner, E.C., Monomorphisms of finitely generated free groups have finitely generated equalizers, Invent. math., 82, 283-289, (1985) · Zbl 0582.20023
[19] Greenberg, L., Commensurable groups of moebius transformations, Ann. of math. stud., 227-237, (1974) · Zbl 0295.20054
[20] Hall, M., A topology for free groups and related groups, Ann. of math. (2), 52, 127-139, (1950) · Zbl 0045.31204
[21] Higman, G., A finitely related group with an isomorphic proper factor group, J. London math. soc., 26, 59-61, (1951) · Zbl 0042.02103
[22] Howson, A.G., On the intersection of finitely generated free groups, J. London math. soc., 29, 428-434, (1954) · Zbl 0056.02106
[23] Kapovich, I., Howson property and one-relator groups, Comm. algebra, 27, 1057-1072, (1999) · Zbl 0922.20046
[24] Kapovich, I., A non-quasiconvexity embedding theorem for hyperbolic groups, Math. proc. Cambridge philos. soc., 127, 461-486, (1999) · Zbl 0942.20026
[25] Kapovich, I., Mapping tori of endomorphisms of free groups, Comm. algebra, 28, 2895-2917, (2000) · Zbl 0953.20035
[26] Kapovich, I.; Weidmann, R., On the structure of two-generated hyperbolic groups, Math. Z., 231, 783-801, (1999) · Zbl 0931.20036
[27] Kapovich, I.; Wise, D.T., The equivalence of some residual properties of word-hyperbolic groups, J. algebra, 223, 562-583, (2000) · Zbl 0951.20029
[28] Karrass, A.; Solitar, D., The free product of two groups with a malnormal amalgamated subgroup, Canad. J. math., 23, 933-959, (1971) · Zbl 0247.20028
[29] Kharlampovich, O.; Myasnikov, A., Hyperbolic groups and free constructions, Trans. amer. math. soc., 350, 571-613, (1998) · Zbl 0902.20018
[30] Kharlampovich, O.; Myasnikov, A., Irreducible affine varieties over a free group. I. irreducibility of quadratic equations and nullstellensatz, J. algebra, 200, 472-516, (1998) · Zbl 0904.20016
[31] Kharlampovich, O.; Myasnikov, A., Irreducible affine varieties over a free group. II. systems in triangular quasi-quadratic form and description of residually free groups, J. algebra, 200, 517-570, (1998) · Zbl 0904.20017
[32] Lyndon, R.C.; Schupp, P.E., Combinatorial group theory, Ergebnisse der Mathematik und ihrer grenzgebiete, 89, (1977), Springer-Verlag Berlin · Zbl 0368.20023
[33] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory, (1976), Dover New York
[34] Neumann, H., On the intersection of finitely generated free groups, Publ. math. debrecen, 4, 186-189, (1956) · Zbl 0070.02001
[35] Neumann, W.D., On intersections of finetly generated subgroups of free groups, Groups—canberra 1989, (1990), Springer-Verlag Berlin, p. 161-170
[36] Nielsen, J., A basis for subgroups of free groups, Math. scand., 3, 31-43, (1955) · Zbl 0066.01202
[37] Nielsen, J., ()
[38] Nielsen, J., ()
[39] Schaufele, Ch.; Zumoff, N., *-groups, graphs, and bases, Topology and combinatorial group theory, (1990), Springer-Verlag Berlin, p. 186-191 · Zbl 0743.20019
[40] Sims, C.C., Computation with finitely presented groups, (1994), Cambridge Univ. Press Cambridge · Zbl 0828.20001
[41] Stallings, J.R., Topology of finite graphs, Invent. math., 71, 551-565, (1983) · Zbl 0521.20013
[42] Stallings, J.R.; Wolf, A.R., The Todd-Coxeter process, using graphs, Combinatorial group theory and topology, (1987), Princeton Univ. Press Princeton, p. 157-161 · Zbl 0619.20014
[43] Stallings, John R., Foldings of G-trees, Arboreal group theory, (1991), Springer-Verlag New York, p. 355-368 · Zbl 0782.20018
[44] Takahasi, M., Note on chain conditions in free groups, Osaka math. J., 3, 221-225, (1951) · Zbl 0044.01106
[45] R. Weidmann, A Grushko theorem for 1-acylindrical splittings, Bochum University, preprint, 1999. · Zbl 0987.20010
[46] D. T. Wise, The residual finiteness of negatively curved polygons of finite groups, Cornell University, preprint, 1997. · Zbl 1040.20024
[47] D. T. Wise, The residual finiteness of positive one-relator groups, Cornell University, preprint, 1998. · Zbl 1023.20011
[48] D. T. Wise, A residually finite version of Rips’ construction, Cornell University, preprint, 1999.
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.