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The groups IA(Fn) and GLn of various polynomial rings

1.
The non-finite presentability of IA(F3) and $GL_2({\bf Z}[t,t^{-1}])$ (with J. McCool), Inventiones Math. 129 (1997) 595-606.

2.
Free quotients of SL2(R[x]) (with J. McCool), Proc. Amer. Math. Soc. 125 (1997) 1585-1588.

3.
A complex for GL2 and its contractibility in the case of Laurent polynomial rings (with J. McCool), in preparation.

4.
Presenting $GL_n(k\langle T\rangle)$ (with J. McCool), Journal of Pure Appl. Algebra 141 (1999) 175-183.



SUMMARY:


The group IA(Fn) is the kernel of the natural homomorphism $Aut(F_n)\to GL_n({\bf Z})$. Magnus showed in [M] that these groups are finitely generated and later asked about their finite presentability. In [1] we prove that IA(F3) is not finitely presentable. The paper utilizes a known homomorphism $IA(F_n)\to
GL_{n-1}({\bf Z}[t,t^{-1}])$, and as a by-product gives non-finite presentability of $GL_2({\bf Z}G)$ for every group that maps homomorphicaly onto the infinite cycle. The heart of the proof is an analysis of relations in the group GL2(k[t,t-1]) where k is a (finite) field. By a theorem of Stuhler [S], these groups are not finitely presentable, but we needed a stronger result. We obtained an alternative proof in [3] by utilizing a complex ${\mit\Delta}(R)$ that we constructed for arbitrary rings R and proved its contractibility when R=k[t,t-1]. The complex ${\mit\Delta}(R)$ is analogous to the unstable Volodin complex (foundations of Algebraic K-theory) and is probably of more general interest. For example, there is an exact sequence $1\to
\pi_1({\mit\Delta}(R))\to GEU_2(R)\to GL_2(R) \to \pi_0({\mit\Delta}(R)) \to 1$, where GEU2stands for Sylvester's ``general elementary universal'' group. Finally, for many rings R, we construct in [2] homomorphisms from SL2(R[x]) onto free groups of infinite rank, improving significantly the results of Grunewald-Mennicke-Vaserstein [GMV]. In [4] we prove non-finite presentability of GL3 and finite presentability of GLn $(n\ge4)$ for large classes of rings, improving results of several authors.



REFERENCES:


[GMV]
F. Grunewald, J. Mennicke and L. Vaserstein, On the groups $SL_2({\bf Z}[x])$ and SL2(k[x,y]), Israel J. Math. 86 (1994), 157-193.

[M]
W. Magnus, Uber n-dimensionale Gittertransformationen, Acta Mathematica 64 (1934), 353-367.

[S]
U. Stuhler, Zur Frage der Endlichen Präsentierbarkeit gewisser arithmetischer Gruppen im Funktionenkörperfall, Math. Ann. 224 (1976), 217-232.


next up previous
Next: Algorithmic and cohomological aspects Aut(Fn) Up: SUMMARY OF RESULTS Previous: G-graphs and automorphisms of

2000-03-16