Transactions of the American Mathematical Society

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

Author(s): George M. Bergman
Journal: Trans. Amer. Math. Soc. 351 (1999), 1531-1550.
MSC (1991): Primary 20E05, 20E06, 20J05; Secondary 05E20, 20C07, 20E08
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Abstract: For

a free group of finite rank, it is shown that the fixed subgroup of any set

of endomorphisms of

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.

1.: George M. Bergman, Embedding rings in completed graded rings. 4. Commutative algebras, J. Alg. 84 (1983), 62-106. MR 85i:16001d
2.: M. Cohen, W. Metzler and B. Zimmermann, What does a basis of look like?, Math. Ann. 257 (1981), 435-445. MR 82m:20028
3.: P. M. Cohn, Skew fields. Theory of general division rings, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, 1995. MR 97d:12003
4.: D. J. Collins and E. C. Turner, All automorphisms of free groups with maximal rank fixed groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), 615-630. MR 96j:20037
5.: Warren Dicks, On equalizers of sections, J. Alg. (to appear).
6.: Warren Dicks and M. J. Dunwoody, Groups acting on graphs, xvi+283pp., Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, 1989. MR 91b:20001
7.: Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, Contemporary Mathematics, vol. 195, 1996, pp. x+81. MR 97h:20030
8.: Herbert Federer and Bjarni Jónsson, Some properties of free groups, Trans. Amer. Math. Soc. 68 (1950), 1-27. MR 11:323a
9.: R. Z. Goldstein and E. C. Turner, Fixed subgroups of homomorphisms of free groups, Bull. London Math. Soc. 18 (1986), 468-470. MR 87m:20096
10.: W. Imrich and E. C. Turner, Endomorphisms of free groups and their fixed points, Math. Proc. Cambridge Phil. Soc. 105 (1989), 421-422. MR 90f:20037
11.: Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Math. u. ihrer Grenzgebiete, vol. 89, Springer-Verlag, 1977. MR 58:28182
12.: Wilhelm Magnus, Abraham Karrass and Donald Solitar, Combinatorial group theory, Interscience Publishers, 1966; Dover Publications, 1976. MR 34:7617; MR 54:10423
13.: A. I. Mal'cev, On the embedding of group algebras in division algebras, (Russian), Doklady Akad. Nauk SSSR 60 (1948), 1499-1501. MR 10:8c
14.: B. H. Neumann, On the number of generators of a free product, J. London Math. Soc. 18 (1943), 12-20. MR 5:58t
15.: Jean-Pierre Serre, Arbres, amalgames, $SL_{2}$ , Astérisque, no.46, 1977, 1983; translated as Trees, Springer-Verlag, 1980. MR 57:16426
16.: John R. Stallings, Topology of finite graphs, Inventiones Math. 71 (1983), 551-565. MR 85m:05037a
17.: Richard G. Swan, Groups of cohomological dimension one, J. Alg. 12 (1969), 585-601. MR 39:1531
18.: Edward C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), 255-263. MR 96m:20039
19.: U. U. Umirbaev, On the ranks of elements of free groups, Fundamental'naya i Prikladnaya Matematika 2 (1996), 313-315 (Russian).

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