Descriptive complexity of subsets of the space of finitely generated groups

Mustafa Gökhan Benli, Burak Kaya

Published 2019-09-24Version 1

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, we establish that the sets of solvable groups and groups of exponential growth are $\mathbf{\Sigma}^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbf{\Pi}^0_2$-complete. This paper is intended to serve as a compilation of results on this theme.