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

Author:

Garreta Albert1,Legarreta Leire2,Miasnikov Alexei3,Ovchinnikov Denis3

Affiliation:

1. Department of Mathematics , University of the Basque Country , Leioa , Spain

2. Department of Mathematics , University of the Basque Country , Bilbao , Spain

3. Department of Mathematical Sciences , Stevens Institute of Technology , Hoboken , NJ 07030 , USA

Abstract

Abstract We study metabelian groups 𝐺 given by full rank finite presentations A R M \langle A\mid R\rangle_{\mathcal{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 | } \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if | R | | A | - 2 \lvert R\rvert\leq\lvert A\rvert-2 , then the Diophantine problem of 𝐺 is undecidable, while it is decidable if | R | | A | \lvert R\rvert\geq\lvert A\rvert . We further prove that if | R | | A | - 1 \lvert R\rvert\leq\lvert A\rvert-1 , then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

Funder

Russian Science Foundation

European Research Council

Eusko Jaurlaritza

Ministerio de Economía, Industria y Competitividad, Gobierno de España

Publisher

Walter de Gruyter GmbH

Subject

Algebra and Number Theory

Reference26 articles.

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4. M. Cordes, M. Duchin, Y. Duong, M.-C. Ho and A. P. Sánchez, Random nilpotent groups I, Int. Math. Res. Not. IMRN 2018 (2018), no. 7, 1921–1953.

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