Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups

Bergman, George

doi:10.1090/S0002-9947-99-02087-5

Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups
HTML articles powered by AMS MathViewer

by George M. Bergman PDF

Trans. Amer. Math. Soc. 351 (1999), 1531-1550 Request permission

Abstract:

For $F$ a free group of finite rank, it is shown that the fixed subgroup of any set $B$ of endomorphisms of $F$ has rank $\leq \operatorname {rank} (F)$, generalizing a recent result of Dicks and Ventura. The proof involves the combinatorics of derivations of groups. Some related questions are examined.

References

George M. Bergman, Embedding rings in completed graded rings. I. Triangular embeddings, J. Algebra 84 (1983), no. 1, 14–24. MR 716768, DOI 10.1016/0021-8693(83)90065-0
M. Cohen, Wolfgang Metzler, and A. Zimmermann, What does a basis of $F(a,\,b)$ look like?, Math. Ann. 257 (1981), no. 4, 435–445. MR 639577, DOI 10.1007/BF01465865
P. M. Cohn, Skew fields, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995. Theory of general division rings. MR 1349108, DOI 10.1017/CBO9781139087193
D. J. Collins and E. C. Turner, All automorphisms of free groups with maximal rank fixed subgroups, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 4, 615–630. MR 1362943, DOI 10.1017/S0305004100074466
Warren Dicks, On equalizers of sections, J. Alg. (to appear).
Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR 1001965
Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, Contemporary Mathematics, vol. 195, American Mathematical Society, Providence, RI, 1996. MR 1385923, DOI 10.1090/conm/195
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Richard Z. Goldstein and Edward C. Turner, Fixed subgroups of homomorphisms of free groups, Bull. London Math. Soc. 18 (1986), no. 5, 468–470. MR 847985, DOI 10.1112/blms/18.5.468
W. Imrich and E. C. Turner, Endomorphisms of free groups and their fixed points, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 3, 421–422. MR 985677, DOI 10.1017/S0305004100077781
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
Jean-Pierre Serre, Arbres, amalgames, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551–565. MR 695906, DOI 10.1007/BF02095993
Richard G. Swan, Groups of cohomological dimension one, J. Algebra 12 (1969), 585–610. MR 240177, DOI 10.1016/0021-8693(69)90030-1
Edward C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), no. 3, 255–263. MR 1374403, DOI 10.1112/blms/28.3.255
U. U. Umirbaev, On the ranks of elements of free groups, Fundamental’naya i Prikladnaya Matematika 2 (1996), 313-315 (Russian).