Anti-Classification Results for Rigidity Conditions in Abelian and Nilpotent Groups

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian groups are complete co-analytic in the space of countable torsion-free abelian groups ($\mathrm{TFAB}_\omega$); (2) the Hopfian groups are complete co-analytic in $\mathrm{TFAB}_\omega$; (3) the co-Hopfian groups are complete co-analytic in the space of countable $2$-nilpotent groups. In combination with our result from [13, S5], which shows that the endo-rigid abelian groups are complete co-analytic in $\mathrm{TFAB}_\omega$, this shows that four major notions of rigidity from (abelian) group theory are as complex as possible as co-analytic problems. Further, the second and third theorem above solve two open questions of Thomas from [18], who asked this for the space of all countable groups. We leave open the question of whether the co-Hopfian mixed abelian groups are complete co-analytic in the space of countable abelian groups, but we reduce the problem to a concrete question on profinite groups, showing that if $G$ is a countable co-Hopfian abelian reduced group, then, for every prime number $p$, the torsion subgroup $\mathrm{Tor}_p(G)$ of $G$ is finite and $G$ embeds in the profinite group $\prod_{p \in \mathbb{P}} \mathrm{Tor}_p(G)$.