- 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 , 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
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
,
and it brings together ideas of combinatorial group theory and
*K*-theory. For example, it is known that the kernel of
is the normal closure of a single
element;
we prove that if
is the quotient of
by the normal
closure of the same element, then the kernel and the cokernel of the
homomorphism
are the unstable Whitehead groups
*Wh*_{2,n}(*G*) and
*Wh*_{1,n}(*G*) respectively. We also prove that if *G* is
finitely presented and ,
then many of the subgroups we study,
including ,
are finitely presented. The case *n*=2 is exceptional,
and in [1] we prove that
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.