Full rank presentations and nilpotent groups: Structure, Diophantine problem, and genericity

Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We study finitely generated nilpotent groups G given by full rank finite presentations 〈A|R〉Nc in the variety Nc of nilpotent groups of class at most c, where c≥2. We prove that if the deficiency |A|−|R| is at least 2 then the group G is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations 〈A|R〉Nc is undecidable if |A|−|R|≥2 and decidable otherwise. We show that this class of groups is rather large since finite presentations asymptotically almost surely have full rank, so a random nilpotent group in the few relators model has a full rank presentation asymptotically almost surely. Full rank presentations give one a useful tool to approach random nilpotent groups and study their properties. Note, that the results above significantly improve our understanding of the Diophantine problem in finitely generated nilpotent groups: from a few special examples of groups with undecidable Diophantine problem we got to the place where we know that the Diophantine problem in all “typical” nilpotent groups is also undecidable.

Original languageEnglish
Pages (from-to)1-34
Number of pages34
JournalJournal of Algebra
Volume556
DOIs
StatePublished - 15 Aug 2020

Keywords

  • Diophantine problem
  • Free nilpotent group
  • Nilpotent group
  • Random group
  • Random walk

Fingerprint

Dive into the research topics of 'Full rank presentations and nilpotent groups: Structure, Diophantine problem, and genericity'. Together they form a unique fingerprint.

Cite this