On the universal theory of torsion and lacunary hyperbolic groups

2009/03/23 by D. Osin, Denis Osin, Osin, D.
Mathematics · #03C60 #20F50 #20F65 #20F67 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO #msc:03C60 #msc:20F50 #msc:20F65 #msc:20F67

paper · pdf · doi:10.48550/arxiv.0903.3978

Some references are added and corrected

openalex publication_date 2009/03/23 · arxiv created 2009/03/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is undecidable. Finally we observe that the universal theory of the class of hyperbolic groups is undecidable and use this observation to construct a lacunary hyperbolic group with undecidable universal theory. Surprisingly, torsion groups play an important role in the proof of the latter results.

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