Effective Subgroup Separability of Finitely Generated Nilpotent Groups

TL;DR

This paper investigates the effective separability of subgroups in finitely generated nilpotent groups, providing polynomial bounds and extending results to virtually nilpotent groups, with exact computations for normal subgroups.

Contribution

It introduces a new function quantifying subgroup separability and establishes polynomial bounds, generalizing previous work and extending findings to virtually nilpotent groups.

Findings

Polynomial upper and lower bounds for subgroup separability

Exact computation for normal subgroups in nilpotent groups

Extension of results to virtually nilpotent groups

Abstract

This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results extend to virtually nilpotent groups.