Residually Finite Algorithmically Finite Groups, Their Subgroups and Direct Products

We construct a finitely generated infinite recursively presented residually finite algorithmically finite group $G$, thus answering a question of Myasnikov and Osin. The group $G$ here is “strongly infinite” and “strongly algorithmically finite”, which means that $G$ contains an infinite Abelian normal subgroup and all finite Cartesian powers of $G$ are algorithmically finite (i.e., for any $n$, there is no algorithm writing out infinitely many pairwise distinct elements of the group $G^n$). We also formulate several open questions concerning this topic.