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Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity

Published online by Cambridge University Press:  20 November 2018

C. K. Gupta ,
N. D. Gupta  and
G. A. Noskov
C. K. Gupta
Affiliation:
University of Manitoba Winnipeg, Manitoba R3T2N2 e-mail:cgupta@ccu.umanitoba.ca
N. D. Gupta
Affiliation:
University of Manitoba Winnipeg, Manitoba R3T2N2 e-mail:ngupta@ccu.umanitoba.ca
G. A. Noskov
Affiliation:
Institute of Information Technology and Applied Mathematics Omsk 644 Russia e-mail:noskov@univer.omsk.su
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Abstract

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For any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

Keywords

20F9920F0520H25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 2 , 01 April 1994 , pp. 298 - 307
Copyright
Copyright © Canadian Mathematical Society 1994

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