Abstract
We examine the automorphism group Aut(F n ) of a free group F n of rank n⩾ 2 on free generators x 1, x 2,...,x n . It is known that Aut(F 2) can be built from cyclic subgroups using a free and semidirect product. A question remains open as to whether this result can be extended to the case n > 2. Every automorphism of Aut(F n ) sending a generator x i to an element f -1 i x π(i) f i , where f i ∈ F n and π is some permutation on a symmetric group S n , is called a conjugating automorphism. The conjugating automorphism group is denoted C n . A set of automorphisms for which π is the identity permutation form a basis-conjugating automorphism group, denoted Cb n . It is proved that Cb n can be factored into a semidirect product of some groups. As a consequence we obtain a normal form for words in C n . For n ⩾ 4, C n and Cb n have an undecidable occurrence problem in finitely generated subgroups. It is also shown that C n , n ⩾ 2, is generated by at most four elements, and we find its respective genetic code, and that Cb n , n⩾ 2, has no proper verbal subgroups of finite width.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
D. Z. Djokovic, “The structure of the automorphism group of a free group on two generators,” Proc. Am. Math. Soc., 88, No. 2, 218–220 (1983).
G. T. Kozlov, “Structure of the group Aut(F 2),” in Algebra, Logics, and Applications [in Russian], Irkutsk University, Irkutsk (1994), pp. 28–32.
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin (1977).
S. Krstic and J. McCool, “The non-finite presentability of IA(F3) and GL2(Z[t, t—1]),” Inv. Math., 129, No. 3, 595–606 (1997).
J. McCool, “On basis-conjugating automorphisms of free groups,” Can. J. Math., 38, No. 6, 1525–1529 (1986).
A. G. Savushkina, “The conjugating automorphism group of a free group,” Mat. Zametki, 60, No. 1, 92–108 (1996).
A. A. Markov, “Fundamentals of algebraic theory of braid groups,” Trudy Mat. Inst. Akad. Nauk SSSR, 16, 1–54 (1945).
J. S. Birman, Braids, Links and Mapping Class Group, Univ. Press, Princeton (1974).
D. J. Collins, and N. D. Gilbert, “Structure and torsion in automorphism groups of free products,” Quart. J. Math. Oxford, 41, No. 162, 155–178 (1990).
The Kourovka Notebook, 15th edn., Institute of Mathematics SO RAN, Novosibirsk (2002).
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], 4th edn., Nauka, Moscow (1996).
H. Servatius, “Automorphisms of graph groups,” J. Alg., 126, No. 1, 34–60 (1989).
H. Servatius, C. Droms, and B. Servatius, “Surface subgroups of graph groups,” Proc. Am. Math. Soc., 106, No. 3, 573–578 (1989).
V. G. Bardakov, “Width of verbal subgroups in some Artin groups, group and metric properties of mappings,” in Collected Papers (in commemoration of Yu. I. Merzlyakov), Novosibirsk University, Novosibirsk (1995), pp. 8–18.
H. S. Coxeter and W. O. Mozer, Generators and Relations for Discrete Groups, Springer, Berlin (1980).
Yu. I. Merzlyakov, Rational Groups [in Russian], 2nd edn., Nauka, Moscow (1987).
V. G. Bardakov, “On a width of verbal subgroups of certain free constructions,” Algebra Logika, 36, No. 5, 494–517 (1997).
A. H. Rhemtulla, “A problem of bounded expressibility in free products,” Proc. Cambridge Phil. Soc., 64, No. 3, 573–584 (1969).
V. G. Bardakov, “Toward a theory of braid groups,” Mat. Sb., 183, No. 6, 3–42 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bardakov, V.G. Structure of a Conjugating Automorphism Group. Algebra and Logic 42, 287–303 (2003). https://doi.org/10.1023/A:1025913505208
Issue Date:
DOI: https://doi.org/10.1023/A:1025913505208