Ordinals arising as residual finiteness depths

Abstract

For every natural number $n$, there exist finitely presented groups with residual finiteness depths $\omega \cdot n$ and $\omega \cdot n+1$. The ordinals that arise as the residual finiteness depth of a finitely generated group (equivalently, a countable group) are $0$, $1$, the countable limit ordinals, and the successors of these limit ordinals.