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Decision problems concerning S-arithmetic groups

Published online by Cambridge University Press:  12 March 2014

Fritz Grunewald and
Daniel Segal
Fritz Grunewald
Affiliation:
Mathematical Institute, University of Bonn, Bonn, West Germany
Daniel Segal
Affiliation:
Mathematical Institute, University of Bonn, Bonn, West Germany
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Extract

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.

We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.

(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases:

= {finitely generated nilpotent groups};

= {(not necessarily associative) rings whose additive group is finitely generated};

= {finitely Z-generated modules over a fixed finitely generated ring}.

Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985, pp. 743 - 772
Copyright
Copyright © Association for Symbolic Logic 1985

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References

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