Reidemeister spectra of free nilpotent groups

Determine the Reidemeister spectrum of the free c-step nilpotent group N_{r,c} for all integers r≥2 and c<2r, thereby completing the cases not covered by the classification of groups with the R∞-property.

Background

The paper studies the Reidemeister spectrum SpecR(N_{r,c}) of the free c-step nilpotent group of rank r. Groups with c≥2r are already classified as having the R∞-property, so the unresolved range is c<2r. Several low-step cases are known, including c=1 and c=2, but a general determination for the remaining parameters is not available. The paper relates this group-theoretic problem to symmetric polynomials and plethysms, but does not solve the full spectrum problem.

References

Determining the Reidemeister spectra of free nilpotent groups and determining the coefficients in the Schur expansion of the plethysms of Schur functions are both well known and well studied open problems in their respective fields. For both problems, significant progress has been made, but complete solutions still remain unknown.

— Reidemeister spectra of free nilpotent groups and plethysms of Schur functions  (2501.02591 - Senden, 5 Jan 2025) in Section 1, subsection “Starting point: Reidemeister spectra of free nilpotent groups”