Scott sentences for certain groups

TL;DR

This paper constructs and analyzes Scott sentences for specific computable groups, demonstrating that these sentences can often be simplified to lower complexity forms, with results extending to finitely generated abelian groups and certain torsion-free groups.

Contribution

It extends the understanding of Scott sentences for computable groups, showing they can often be simplified to computable $d$-$\Sigma_2$ forms, including all finitely generated abelian groups and the infinite dihedral group.

Findings

All studied computable finitely generated groups have computable $\Sigma_3$ Scott sentences.

Some groups admit simpler computable $d$-$\Sigma_2$ Scott sentences.

For certain rank 1 torsion-free abelian groups, the $\Sigma_3$ Scott sentence is optimal.

Abstract

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable $\Sigma_3$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable $d$-$\Sigma_2$" (the conjunction of a computable $\Sigma_2$ sentence and a computable $\Pi_2$ sentence). This was already shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank $1$. These are exactly the additive subgroups of $\mathbb{Q}$. We show that for some of these groups, the computable $\Sigma_3$ Scott sentence is best possible, while for others, there is a computable $d$-$\Sigma_2$ Scott sentence.