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Basis-conjugating automorphisms of a free group and associated Lie algebras

F R Cohen, J Pakianathan, V V Vershinin and Jie Wu

Geometry & Topology Monographs 13 (2008) 147–168

DOI: 10.2140/gtm.2008.13.147

arXiv: math.AT/0610946


Let Fn = ⟨x1, …, xn⟩ denote the free group with generators {x1, …, xn}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of Fn to the general linear group over the integers. In particular among them are the automorphisms χk,i which conjugate the generator xk by the generator xi leaving the xj fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by χk,i for i < k is given here. Partial results are obtained for the group generated by all χk,i.


automorphism group, free group, basis-conjugating automorphism, Lie algebra, cohomology

Mathematical Subject Classification

Primary: 20F28

Secondary: 20F40, 20J06


Received: 31 May 2006
Revised: 6 March 2007
Accepted: 13 March 2007
Published: 22 February 2008

F R Cohen
Department of Mathematics
University of Rochester
Rochester, NY 14627
J Pakianathan
Department of Mathematics
University of Rochester
Rochester, NY 14627
V V Vershinin
Département des Sciences Mathématiques
Université Montpellier II
Place Eugéne Bataillon
34095 Montpellier cedex 5
Sobolev Institute of Mathematics
Novosibirsk, 630090
Jie Wu
Department of Mathematics
National University of Singapore
Singapore 117543
Republic of Singapore