Structure of a~Conjugating Automorphism Group

V. G. Bardakov

Algebra i logika, Tome 42 (2003) no. 5, pp. 515-541.

Résumé

We examine the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$ of rank $n\geqslant 2$ on free generators $x_1,x_2,\ldots,x_n$. It is known that ${\rm Aut}(F_2)$ 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 ${\rm Aut}(F_n)$ sending a generator $x_i$ to an element $f_i^{-1}x_{\pi(i)}f_i$, where $f_i\in F_n$ and $\pi$ 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 $\pi$ 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 $n\geqslant 4$, $C_n$ and $Cb_n$ have an undecidable occurrence problem in finitely generated subgroups. It is also shown that $C_n$, $n\geqslant 2$, is generated by at most four elements, and we find its respective genetic code, and that $Cb_n$, $n\geqslant 2$, has no proper verbal subgroups of finite width.

Détail

Keywords: conjugating automorphism group, basis-conjugating automorphism group, occurrence problem in finitely generated subgroups, factorization of a group into a semidirect product.

Bibliographie

[1] D. Z. Djokovic, “The structure of the automorphism group of a free group on two generators”, Proc. Am. Math. Soc., 88:2 (1983), 218–220 | DOI | MR | Zbl

[2] G. T. Kozlov, “Stroenie gruppy $\operatorname{Aut}(F_2)$”, Algebra, logika i prilozheniya, Irkutskii gos. un-t, Irkutsk, 1994, 28–32

[3] R. Lindon, P. Shupp, Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[4] S. Krstic, J. McCool, “The non-finite presentability of $IA(F_3)$ and $GL_2(\mathbb{Z}|t,t^{-1}|)$”, Invent. Math., 129:3 (1997), 595–606 | DOI | MR | Zbl

[5] J. McCool, “On basis-conjugating automorphisms of free groups”, Can. J. Math., 38:6 (1986), 1525–1529 | MR | Zbl

[6] A. G. Savushkina, “O gruppe sopryagayuschikh avtomorfizmov svobodnoi gruppy”, Matem. zametki, 60:1 (1996), 92–108 | MR | Zbl

[7] A. A. Markov, “Osnovy algebraicheskoi teorii kos”, Tr. Matem. in-ta AN, 16 (1945), 1–54 | MR | Zbl

[8] J. S. Birman, Braids, links and mapping class group, Univ. press, Princeton-Tokyo, 1974 | MR | Zbl

[9] D. J. Collins, N. D. Gilbert, “Structure and torsion in automorphism groups of free products”, Quart. J. Math. Oxford, 41:162 (1990), 155–178 | DOI | MR | Zbl

[10] Nereshennye voprosy teorii grupp. Kourovskaya tetrad, 15-e izd., In-t matem. SO RAN, Novosibirsk, 2002 | MR

[11] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, 4-e izd., Nauka, M., 1996 | MR | Zbl

[12] H. Servatius, “Automorphisms of graph groups”, J. Algebra, 126:1 (1989), 34–60 | DOI | MR | Zbl

[13] H. Servatius, C. Droms, B. Servatius, “Surface subgroups of graph groups”, Proc. Am. Math. Soc., 106:3 (1989), 573–578 | DOI | MR | Zbl

[14] V. G. Bardakov, “Shirina verbalnykh podgrupp nekotorykh grupp Artina, gruppovye i metricheskie svoistva otobrazhenii”, sb. rabot, posv. pamyati Yu. I. Merzlyakova, Novosibirskii gos. un-t, Novosibirsk, 1995, 8–18 | MR | Zbl

[15] G. S. Kokseter, U. O. Mozer, Porozhdayuschie elementy i opredelyayuschie sootnosheniya diskretnykh grupp, Nauka, M., 1980

[16] Yu. I. Merzlyakov, Ratsionalnye gruppy, 2-e izd., Nauka, M., 1987 | MR | Zbl

[17] V. G. Bardakov, “O shirine verbalnykh podgrupp nekotorykh svobodnykh konstruktsii”, Algebra i logika, 36:5 (1997), 494–517 | MR | Zbl

[18] A. H. Rhemtulla, “A problem of bounded expressibility in free products”, Proc. Camb. Philos. Soc., 64:3 (1969), 573–584 | DOI | MR

[19] V. G. Bardakov, “K teorii grupp kos”, Matem. sb., 183:6 (1992), 3–42 | Zbl