Equations in nilpotent groups

2014/01/10 by Moon Duchin, Hao Liang, Duchin, Moon +3 · 3 citations
Mathematics · #20F10 #20F18 #20F70 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:20F10 #msc:20F18 #msc:20F70

paper · pdf · doi:10.48550/arxiv.1401.2471

arxiv created 2014/01/10 · openalex publication_date 2014/01/10 · arxiv updated 2014/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

Cited by

• The Diophantine problem in the Baumslag-Gersten group
• Decidability of equations and first-order theory in Seifert 3-manifold groups
• One variable equations over the lamplighter group

Related

• The complexity of the equation solvability problem over semipattern\n groups
• Quadratic Diophantine equations, the Heisenberg group and formal languages
• Some notes on extended equation solvability and identity checking for\n groups
• Infinite systems of equations in abelian and nilpotent groups
• Diophantine problems in solvable groups