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arxiv: 2504.03596 · v3 · submitted 2025-04-04 · 🧮 math.GR

Algorithms for twisted conjugacy classes of polycyclic-by-finite groups II

Sam Tertooy This is my paper

Pith reviewed 2026-05-22 21:17 UTC · model grok-4.3

classification 🧮 math.GR
keywords twisted conjugacy classesReidemeister numberpolycyclic-by-finite groupsalgorithmsdouble cosetsaffine actionsgroup homomorphisms
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The pith

An algorithm decides whether the Reidemeister number is finite for homomorphisms between polycyclic-by-finite groups and returns representatives of the twisted conjugacy classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an algorithm that processes a pair of homomorphisms between polycyclic-by-finite groups. It determines if the associated Reidemeister number, which counts the twisted conjugacy classes, is finite. When the number is finite, the algorithm produces a set of representatives for those classes. The method further applies to finding double cosets and orbits in affine group actions. Readers interested in computational aspects of group theory would find this useful for turning theoretical questions about classes into explicit lists.

Core claim

We construct an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite, and if so returns a set of representatives of the twisted conjugacy classes. Moreover, we show how this algorithm can be applied to compute double cosets and orbits of affine actions.

What carries the argument

The algorithm for deciding finiteness of the Reidemeister number and enumerating representatives, built on the structural properties of polycyclic-by-finite groups that make subgroup problems and conjugacy decidable.

If this is right

  • The Reidemeister number can be tested for finiteness by an explicit procedure.
  • A finite set of representatives for twisted conjugacy classes can be computed.
  • Double cosets between subgroups can be listed using the same technique.
  • Orbits of affine actions on the groups become computable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Implementation in computer algebra systems would allow routine calculation of these invariants for concrete groups.
  • The technique might adapt to other group classes that share decidable conjugacy and membership problems.
  • Applications could extend to topological questions involving fixed points of maps on spaces with fundamental groups of this type.

Load-bearing premise

The input groups are polycyclic-by-finite so that their word problem, conjugacy problem, and subgroup lattice properties remain computable.

What would settle it

Implement the algorithm for a pair of concrete homomorphisms on small polycyclic groups like the Heisenberg group over integers, compute the twisted classes by hand, and verify if the output set matches in size and content.

read the original abstract

We construct an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite, and if so returns a set of representatives of the twisted conjugacy classes. Moreover, we show how this algorithm can be applied to compute double cosets and orbits of affine actions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper constructs an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite and, if so, returns a set of representatives of the twisted conjugacy classes. It further shows applications of this algorithm to computing double cosets and orbits of affine actions.

Significance. If the algorithm is correct and terminates, the result supplies an explicit decision procedure for twisted conjugacy finiteness in a class of groups already known to have decidable word problem, subgroup membership, and centralizer computations. This extends the algorithmic toolkit for polycyclic-by-finite groups and provides concrete methods for double-coset and orbit problems that arise in several areas of group theory.

minor comments (2)
  1. [§3] §3: the termination argument for the main search procedure relies on the finiteness of certain centralizers; a brief remark on how the polycyclic presentation is used to enumerate them would improve readability.
  2. [§5] §5, application to double cosets: the reduction is stated clearly but the input encoding (how the coset representatives are represented) is not made explicit; adding one sentence on data structures would help implementers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper and for recommending minor revision. The report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity in algorithmic derivation

full rationale

The paper constructs an explicit algorithm that reduces the decision problem for finiteness of the Reidemeister number (and computation of representatives) to a finite sequence of standard decidable operations on polycyclic-by-finite groups, including word-problem solvability, subgroup membership, and centralizer computations. These properties are external, well-established facts about the input class and are not derived from or fitted to the algorithm's own outputs. No equations or steps equate the claimed result to a parameter fitted from the same data, a self-citation chain that defines the result by construction, or an ansatz smuggled in via prior work by the same authors. The derivation is therefore self-contained and independent.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The algorithm depends on the structural and algorithmic properties that define polycyclic-by-finite groups; no new entities are postulated and no numerical parameters are fitted.

axioms (1)
  • domain assumption Polycyclic-by-finite groups admit algorithms for subgroup membership, finite-index computations, and related decision problems.
    This is the standard background fact that makes algorithmic work on this class feasible; it is invoked implicitly by the claim that an algorithm exists.

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Reference graph

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