Algorithms for twisted conjugacy classes of polycyclic-by-finite groups II
Sam Tertooy
TL;DR
The paper presents a comprehensive suite of algorithms to compute Reidemeister numbers and representatives for twisted conjugacy classes in polycyclic-by-finite groups, generalizing from the case $G=H$ to $G$-and-$H$ pairs and incorporating a normal subgroup $N$ via $R_N(\varphi,\psi)$. It builds a layered framework—starting with Coin$_N(\varphi,\psi)$ and standard quintuples, then advancing through nilpotent-by-finite, metabelian, ABCD, and finally general nilpotent-by-abelian cases—each supported by new auxiliary reduction lemmas and byfinite techniques. The work unifies and extends algorithmic twisted-conjugacy theory, enabling practical computation of finiteness, orbit representatives, double cosets, and affine-action dynamics, with implementations in the GAP TwistedConjugacy package. This advances both the theoretical understanding and the computational toolkit for Reidemeister theory in a broad, practically relevant class of groups, with direct applications to topological coincidence theory and related algebraic problems.
Abstract
We construct an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite, and additionally returns a set of representatives of the twisted conjugacy classes if it is. Moreover, we show how this algorithm can be applied to compute double cosets and orbits of affine actions.