On groups and fields interpretable in torsion-free hyperbolic groups

2012/10/21 by Chloé Perin, Anand Pillay, Perin, Chloé +5
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO

paper · pdf · doi:10.48550/arxiv.1210.5757

12 pages

arxiv created 2013/02/19 · arxiv updated 2013/02/20

Abstract

We prove that the generic type of a non-cyclic torsion-free hyperbolic group G is foreign to any interpretable abelian group, hence also to any interpretable field. This result depends, among other things, on the definable simplicity of a non-cyclic torsion-free hyperbolic group, and we take the opportunity to give a proof of the latter using Sela's description of imaginaries in torsion-free hyperbolic groups. We also use the description of imaginaries to prove that if F is a free group of rank > 2 then no orbit of a finite tuple from F under Aut(F) is definable.

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