Geodesic
Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity
Canadian journal of mathematics, Tome 46 (1994) no. 2, pp. 298-307

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1994-014-6
Mots-clés : 20F99, 20F05, 20H25 
Gupta, C. K.; Gupta, N. D.; Noskov, G. A. Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity. Canadian journal of mathematics, Tome 46 (1994) no. 2, pp. 298-307. doi: 10.4153/CJM-1994-014-6
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