Finitely Generated Groups Are Universal

TL;DR

This paper demonstrates that finitely generated groups can simulate any finitely generated structure's computability properties, establishing their universality within the constraints of finite generation.

Contribution

It proves finitely generated groups are as universal as possible for finitely generated structures, contrasting with finitely generated fields.

Findings

Finitely generated groups can replicate any finitely generated structure's computability properties.

Finiteness constraints limit the universality of finitely generated fields.

Results impact the understanding of quasi Scott sentences in computable structure theory.

Abstract

Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences.