$X$-torsion and universal groups

Abstract

For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$ to be all elements $g\in G$ with $g^n=e$ for some $n\in X$. With $X$ recursively enumerable, we give two independent proofs (group-theoretic, and model-theoretic) that there exists a universal finitely presented $X$-torsion-free group; one which contains all finitely presented $X$-torsion-free groups. We also show that, if $X$ is recursively enumerable, then the set of finite presentations of $X$-torsion-free groups is ${\Pi }_2^0$-complete in Kleene's arithmetic hierarchy.

10 pages. This is the first version, comments are welcome

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