Generically Computable Abelian Groups

Abstract

Generically computable sets, as introduced by Jockusch and Schupp, have been of great interest in recent years. This idea of approximate computability was motivated by asymptotic density problems studied by Gromov in combinatorial group theory. More recently, we have defined notions of generically computable structures, and studied in particular equivalence structures and injection structures. A structure is said to be generically computable if there is a computable substructure defined on an asymptotically dense set, where the functions are computable and the relations are computably enumerable. It turned out that every equivalence structure has a generically computable copy, whereas there is a non-trivial characterization of the injection structures with generically computable copies.

In this paper, we return to group theory, as we explore the generic computablity of Abelian groups. We show that any Abelian p-group has a generically computable copy and that such a group has a $\varSigma _2$-generically computably enumerable copy if and only it has a computable copy. We also give a partial characterization of the $\varSigma _1$-generically computably enumerable Abelian p-groups. We also give a non-trivial characterization of the generically computable Abelian groups that are not p-groups.

This research was partially supported by the National Science Foundation SEALS grant DMS-1362273. The work was done partially while the latter two authors were visiting the Institute for Mathematical Sciences, National University of Singapore, in 2017. The visits were supported by the Institute. This material is partially based upon work supported by the National Science Foundation under grant DMS-1928930 while all three authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2020 semester. Harizanov was partially supported by the Simons Foundation grant 853762 and NSF grant DMS-2152095.