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Automorphisms of free G-groups

1.
The non-finite presentability of the free Z-group of rank 2 (with J. McCool), J. London Math. Soc. 56 (1997) 254-274.

2.
Finite presentability of $\Phi_n(G)$, $GL_n({\bf Z}G)$ and their elementary subgroups and Steinberg groups (with J. Kiralis and J. McCool), Proc. London Math. Soc. 73 (1996), 575-622.



SUMMARY:


A free G-group is a non-abelian analogue of a free G-module; it is a free group with G acting freely on a set of free generators. The equivariant automorphism group $\Phi_n(G)$ of the free G-group of rank n occurred in Kiralis's study of ``non-abelian K-theory'' [K]. It was independently introduced also by Denk-Metzler [DM] and McCool [M] who gave an infinite presentation of it. The paper [2] contains an extensive study of the natural homomorphism $\Phi_n(G)\to GL_n({\bf Z})$, and it brings together ideas of combinatorial group theory and K-theory. For example, it is known that the kernel of $Aut(F_n)=\Phi_n(1)\to GL_n({\bf Z})$ is the normal closure of a single element; we prove that if $\Psi_n(G)$ is the quotient of $\Phi_n(G)$ by the normal closure of the same element, then the kernel and the cokernel of the homomorphism $\Psi_n(G)\to GL_n({\bf Z})$ are the unstable Whitehead groups Wh2,n(G) and Wh1,n(G) respectively. We also prove that if G is finitely presented and $n\ge3$, then many of the subgroups we study, including $\Phi_n(G)$, are finitely presented. The case n=2 is exceptional, and in [1] we prove that $\Phi_2(G)$ is not finitely presentable whenever the abelianization G/[G,G] is infinite.



REFERENCES:


[DM]
G. Denk and W. Metzler, Nielsen reduction in free groups with operators, Fund. Math. 129 (1988), 181-197.

[K]
J. Kiralis, Nonabelian K-theory and pseudo-isotopies of 3-manifolds, K-theory (to appear).

[M]
J. McCool, The automorphism groups of free G-groups, J. Algebra 166 (1994), 158-180.


next up previous
Next: Kervaire-Laudenbach Conjecture Up: SUMMARY OF RESULTS Previous: Algorithmic and cohomological aspects Aut(Fn)

2000-03-16