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.
1. G. N. Arzhantseva and A. Y. Ol’shanskiĭ,
The class of groups all of whose subgroups with lesser number of generators are free is generic,
Math. Notes 59 (1996), no. 4, 350–355. 2. S. Burris and H. P. Sankappanavar,
A Course in Universal Algebra,
Grad. Texts in Math. 78,
Springer, New York, 1981. 3. C. Champetier,
Propriétés statistiques des groupes de présentation finie,
Adv. Math. 116 (1995), no. 2, 197–262. 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. 5. M. Davis, H. Putnam and J. Robinson,
The decision problem for exponential diophantine equations,
Ann. of Math. (2) 74 (1961), 425–436.
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