Descriptive complexity of subsets of the space of finitely generated groups

TL;DR

This paper analyzes the descriptive complexity of various subsets of finitely generated groups within the space of marked groups, classifying them into specific Borel hierarchy levels using advanced group-theoretic constructions.

Contribution

It provides the first comprehensive classification of the descriptive complexity of key group properties in the space of finitely generated groups.

Findings

Solvable groups are $oldsymbol{ ext{Σ}}^0_2$-complete.

Groups of exponential growth are $oldsymbol{ ext{Σ}}^0_2$-complete.

Groups with decidable word problem are $oldsymbol{ ext{Σ}}^0_2$-complete.

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 $\mathbf{\Sigma}^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbf{\Pi}^0_2$-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.