Ordinals arising as residual finiteness depths

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.