# Effective Twisted Conjugacy Separability of Nilpotent Groups

## TL;DR

This paper explores the effective twisted conjugacy separability in finitely generated nilpotent groups, establishing polynomial bounds and extending results to virtually nilpotent groups, with precise calculations for certain classes.

## Contribution

It introduces the concept of effective twisted conjugacy separability, proves polynomial bounds for nilpotent groups, and extends conjugacy separability results to virtually nilpotent groups.

## Key findings

- Polynomial upper bounds for twisted conjugacy separability in nilpotent groups
- Extension of conjugacy separability to virtually nilpotent groups
- Precise calculation of conjugacy separability for nilpotent groups of class 2

## Abstract

This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.