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THE NON-FINITE PRESENTABILITY OF THE AUTOMORPHISM GROUP OF THE FREE Z-GROUP OF RANK TWO

Published online by Cambridge University Press:  01 October 1997

SAVA KRSTIC  and
JAMES McCOOL
SAVA KRSTIC
Affiliation:
Department of Mathematics, Tufts University, Medford, Massachusetts 02155, USA. E-mail: [email protected]
JAMES McCOOL
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1. E-mail: [email protected]
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Abstract

Let G be a group, and let Fn[G] be the free G-group of rank n. Then Fn[G] is just the natural non-abelian analogue of the free ℤG-module of rank n, and correspondingly the group Φn(G) of equivariant automorphisms of Fn[G] is a natural analogue of the general linear group GLn(ℤG). The groups Φn(G) have been studied recently in [4, 8, 5]. In particular, in [5] it was shown that if G is not finitely presentable (f.p.) then neither is Φn(G), and conversely, that Φn(G) is f.p. if G is f.p. and n≠2. It is a common phenomenon that for small ranks the automorphism groups of free objects may behave unstably (see the survey article [2]), and the main aim of the present paper is to show that this turns out to be the case for the groups Φ2(G).

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 264 - 274
Copyright
The London Mathematical Society 1997

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