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Free quotients of infinite rank of GL2 over Dedekind domains

Published online by Cambridge University Press:  14 November 2011

A. W. Mason
A. W. Mason
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK (awm@maths.gla.ac.uk)
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Extract

This paper is concerned with integral domains R, for which the factor group SL2(R)/U2(R) has a non-trivial, free quotient, where U2(R) is the subgroup of GL2(R) generated by the unipotent matrices. Recently, Krstić and McCool have proved that SL2(P[x])/U2(P[x]) has a free quotient of infinite rank, where P is a domain which is not a field. This extends earlier results of Grunewald, Mennicke and Vaserstein.

Any ring of the type P[x] has Krull dimension at least 2. The purpose of this paper is to show that result of Krstić and McCool extends to some domains of Krull dimension 1, in particular to certain Dedekind domains. This result, which represents a two-dimensional anomaly is the best possible in the following sense. It is well known that SL2(R) = U2(R), when R is a domain of Krull dimension zero, i.e. when R is a field. It is already known that for some arithmetic Dedekind domains A, the factor group SL2(A)/U2(A) has a free quotient of finite (and not infinite) rank.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 77 - 84
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Grunewald, F., Mennicke, J. and Vaserstein, L.. On the groups SL 2(ℤ[x]) and SL 2(k[x, y]). Israel J. Math. 86 (1994), 157–193.CrossRefGoogle Scholar
2Hahn, A. J. and O'Meara, O. T.. The classical groups and K-theory (Berlin, Heidelberg: Springer, 1989).CrossRefGoogle Scholar
3Krstić, S. and McCool, J.. Free quotients of SL 2(R[x]). Proc. Am. Math. Soc. 125 (1997), 1585–1588.CrossRefGoogle Scholar
4Liehl, B.. On the group SL 2 over orders of arithmetic type. J. Reine Angew. Math. 323 (1981), 153–171.Google Scholar
5Magnus, W., Karrass, A. and Solitar, D.. Combinatorial group theory (New York: Wiley, 1966).Google Scholar
6Neumann, P. M.. The SQ-universality of some finitely presented groups. J. Austral. Math. Soc. 16 (1973), 1–6.CrossRefGoogle Scholar
7Pride, S. J.. The concept of ‘largeness’ in group theory. Word problems II. In The Oxford book (ed. Adjan, S. J., Boone, W. W. and Higman, G.), pp. 299–335 (Amsterdam: North-Holland, 1980).Google Scholar
8Serre, J.-P.. Trees (Berlin, Heidelberg: Springer, 1980).CrossRefGoogle Scholar
9Stichtenoth, H.. Algebraic function fields and codes (Berlin, Heidelberg: Springer, 1993).Google Scholar
10Vaserstein, L. N.. On the group SL 2 over Dedekind rings of arithmetic type. Math. USSR Sb 18 (1972), 321–332.CrossRefGoogle Scholar