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On certain questions of the free group automorphisms theory. (English) Zbl 1147.20032

The authors consider certain subgroups of the automorphism group \(\operatorname{Aut}(F_n)\) of the free group \(F_n\) and study them as far as their presentation and their linearity is concerned. In particular, they consider the group \(\text{IA}(F_n)\) of IA-automorphisms of \(F_n\) (automorphisms which induce the identity of \(F_n/F_n'\)), the group \(Cb_n\) of basis conjugating automorphisms of \(F_n\), the braid group \(B_n\) as a subgroup of \(\operatorname{Aut}(F_n)\) and other subgroups. If the generators of \(F_n\) are denoted by \(x_1,x_2,\dots,x_n\), the subgroup \(Cb^+_n\) of \(Cb_n\) for \(n\geq 2\) is generated by \(\varepsilon_{ij}\), \(i>j\) where \(\varepsilon_{ij}\colon x_i\mapsto x^{-1}_jx_ix_j\), \(i\neq j\), \(x_i\mapsto x_k\), \(k\neq i\). The braid group generated by the \(\sigma_i\)’s is mapped homomorphically on the permutation group \(S_n\), by sending \(\sigma_i\) to the transposition \((i,i+1)\), \(i=1,\dots,n-1\). The kernel of this mapping is the ‘pure braid group’ \(P_n\).
The authors exhibit various known and new properties of these groups. Furthermore they prove that the groups \(\text{IA}(F_n)\) are not linear for all \(n\geq 3\). A. Pettet in 2006 [Cohomology of some subgroups of the automorphism group of a free group. Ph.D. Thesis. Univ. Chicago] proved the same result for \(n\geq 5\). They also prove, using Fox differential calculus, that \(Cb^+_4\) and \(P_4\) are not isomorphic. They finish by stating some open questions.

MSC:

20F28 Automorphism groups of groups
20E05 Free nonabelian groups

References:

[1] DOI: 10.1112/plms/s3-15.1.239 · Zbl 0135.04502 · doi:10.1112/plms/s3-15.1.239
[2] Bardakov V., Algebra i Logik 42 pp 515– (2003)
[3] Bardakov V., Sib. Mat. J. 46 pp 17– (2005)
[4] DOI: 10.1090/S0894-0347-00-00361-1 · Zbl 0988.20021 · doi:10.1090/S0894-0347-00-00361-1
[5] Birman J., Braids, Links and Mapping Class Group (1974)
[6] DOI: 10.2140/agt.2001.1.445 · Zbl 0977.57014 · doi:10.2140/agt.2001.1.445
[7] Crowell R., Introduction to Knot Theory (1963)
[8] DOI: 10.1007/BF01394780 · Zbl 0574.55010 · doi:10.1007/BF01394780
[9] DOI: 10.1016/0021-8693(92)90029-L · Zbl 0780.20023 · doi:10.1016/0021-8693(92)90029-L
[10] DOI: 10.2307/3062152 · Zbl 1020.20025 · doi:10.2307/3062152
[11] Lyndon R., Combinatorial Group Theory (1977) · Zbl 0368.20023
[12] DOI: 10.4153/CJM-1986-073-3 · Zbl 0613.20024 · doi:10.4153/CJM-1986-073-3
[13] Savushkina A., Mat. Zametki 60 pp 92– (1996)
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