Metabelian groups: full-rank presentations, randomness and Diophantine problems

2020/06/11 by Garreta, Albert, Legarreta, Leire, Miasnikov, Alexei +1 · 1 citation
#03B25 #03D35 #20F05 #20F10 #20F16 #20F69 #20F70 #60G99 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2006.06371

Abstract

We study metabelian groups G given by full rank finite presentations ⟨ A | R ⟩_M in the variety M of metabelian groups. We prove that G is a product of a free metabelian subgroup of rank max\0, |A|-|R|\ and a virtually abelian normal subgroup, and that if |R| ≤ |A|-2 then the Diophantine problem of G is undecidable, while it is decidable if |R|≥ |A|. We further prove that if |R| ≤ |A|-1 then in any direct decomposition of G all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

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• The Diophantine problem in the Baumslag-Gersten group

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