Ordinals arising as residual finiteness depths

TL;DR

This paper constructs finitely presented groups with specific residual finiteness depths corresponding to certain ordinals, revealing the possible ordinal values for these depths.

Contribution

It demonstrates the existence of finitely presented groups with residual finiteness depths at specific ordinal levels, expanding understanding of the ordinal spectrum in group theory.

Findings

Existence of groups with residual finiteness depths $\, ext{omega} imes n$ and $ ext{omega} imes n + 1$

Residual finiteness depths are limited to certain ordinals: 0, 1, countable limit ordinals, and their successors

Characterization of the ordinals arising as residual finiteness depths in finitely generated groups

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.