Descriptive complexity of subsets of the space of finitely generated groups

2019/09/24 by Benli, Mustafa Gökhan, Kaya, Burak
#03E15 #20F65 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.1909.11163

Abstract

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are \mathbfΣ⁰₂-complete and that the sets of periodic groups and groups of intermediate growth are \mathbfΠ⁰₂-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as a compilation of results on this theme.

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