The product formula for Reidemeister numbers on nilpotent groups

TL;DR

This paper extends the product formula for Reidemeister numbers to central extensions of finitely generated torsion-free nilpotent groups and develops tools to analyze their Reidemeister spectra.

Contribution

It generalizes the product formula to central extensions and introduces methods to compute Reidemeister numbers and spectra for finitely generated nilpotent groups.

Findings

Extended product formula to central extensions.

Provided criteria for infinite Reidemeister numbers.

Constructed groups with full Reidemeister spectrum.

Abstract

We study the product formula for Reidemeister numbers on finitely generated torsion-free nilpotent groups in two ways. On the one hand, we generalise the product formula to central extensions. On the other hand, we derive general results for finitely generated (torsion-free) nilpotent groups from the product formula: we provide a strong tool to prove that an endomorphism on a finitely generated nilpotent group has infinite Reidemeister number and compute Reidemeister numbers on finite index subgroups of finitely generated torsion-free nilpotent groups. Using the latter, we provide an infinite family of groups with full Reidemeister spectrum.