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The computability of group constructions II

Published online by Cambridge University Press:  17 April 2009

R.W. Gatterdam
R.W. Gatterdam
Affiliation:
Department of Mathematics, University of Wisconsin – Parkside, Kenosha, Wisconsin, USA.
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Abstract

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Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En.

It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 1 , February 1973 , pp. 27 - 60
Copyright
Copyright © Australian Mathematical Society 1973

References

[0]Cannonito, F.B. and Gatterdam, R.W., “The computability of group constructions, Part I”, Word problems (Proceedings of the Irvine Conference on Decision Problems in Group Theory. Horth Holland, Amsterdam, in press).Google Scholar
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