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arxiv: 2605.23798 · v1 · pith:FPLAGONWnew · submitted 2026-05-22 · 🧮 math.GR

Algorithms for experimenting with Zariski dense matrix groups over number fields

A. S. Detinko , D. L. Flannery , A. Hulpke This is my paper

Pith reviewed 2026-05-25 02:35 UTC · model grok-4.3

classification 🧮 math.GR
keywords Zariski dense groupscongruence quotientsstrong approximationmatrix groupsnumber fieldscomputational group theoryBianchi groups
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The pith

Algorithms compute the set of congruence quotients for finitely generated Zariski dense subgroups of SL(n, number fields) with n prime.

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

The paper develops algorithms that take a finitely generated Zariski dense group H inside SL(n, P), where P is a number field and n is prime, and output the complete collection of its congruence quotients modulo every maximal ideal of a suitable subring R. This turns the existence statement of the strong approximation theorem into an explicit computational procedure. A sympathetic reader would care because the quotients encode the arithmetic structure of H, and having them listed allows concrete experiments rather than only theoretical guarantees. The algorithms are realized in GAP and tested on degree-two cases with emphasis on Bianchi groups.

Core claim

We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense groups H ≤ SL(n, P), n prime. That is, we present algorithms to find the set of congruence quotients of H modulo all maximal ideals of a finitely generated subring R of P such that H ≤ SL(n, R). The algorithms have been implemented in GAP.

What carries the argument

The algorithms that enumerate congruence quotients of H by using the Zariski density assumption to locate all relevant maximal ideals of R.

If this is right

  • Explicit lists of all congruence images become available for any such H that satisfies the input conditions.
  • Arithmetic properties of the groups can be checked directly through their computed quotients.
  • Systematic experiments on Bianchi groups and other low-degree examples are now feasible.
  • The methods supply a practical tool for exploring the finite images guaranteed by strong approximation.

Where Pith is reading between the lines

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

  • Similar algorithmic ideas could be tested on matrix groups over rings other than number fields if density conditions can be formulated.
  • The computed quotient sets might serve as invariants to distinguish or classify different Zariski dense groups.
  • Effective bounds on the size of the maximal ideals needed could be derived from running the algorithms on families of examples.

Load-bearing premise

The input groups must be finitely generated and Zariski dense in SL(n, P) with n prime; without this density the algorithms may miss or misidentify the full set of congruence quotients.

What would settle it

A concrete Zariski dense finitely generated H in SL(n, P) for which the algorithms produce a set of quotients that omits at least one congruence image known to exist from the strong approximation theorem.

read the original abstract

Let $\mathbb{P}$ be an algebraic number field. We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense groups $H\leq \mathrm{SL}(n,\mathbb{P})$, $n$ prime. That is, we present algorithms to find the set of congruence quotients of $H$ modulo all maximal ideals of a finitely generated subring $R$ of $\mathbb{P}$ such that $H\leq \mathrm{SL}(n,R)$. The algorithms have been implemented in GAP. Potential applications are illustrated by a range of experiments in degree $2$, with a special focus on Bianchi groups.

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 claims to provide algorithms that compute the set of all congruence quotients H mod m, for maximal ideals m of a finitely generated subring R of a number field P, where H is a finitely generated Zariski-dense subgroup of SL(n, P) with n prime; the algorithms are implemented in GAP and demonstrated via experiments in degree 2, with emphasis on Bianchi groups, as a computational analog of the strong approximation theorem.

Significance. If the algorithms correctly realize the claimed computational analog under the stated hypotheses, the work supplies a practical tool for systematically exploring congruence images of arithmetic groups over number fields. The GAP implementation and concrete experiments on Bianchi groups constitute a reproducible contribution that could support further computational investigations in arithmetic group theory.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'a range of experiments' is vague; a brief enumeration of the specific groups or degrees tested would better indicate the scope of the validation.
  2. The manuscript would benefit from an explicit statement, early in the text, of the precise input format expected by the GAP routines (generators, ring presentation, etc.).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; algorithmic claim is self-contained

full rationale

The paper describes algorithms (implemented in GAP) that compute congruence quotients for finitely generated Zariski-dense subgroups H of SL(n,P) with n prime, using the strong approximation theorem as an external mathematical fact rather than deriving it. No equations, fitted parameters, or derivations appear that reduce by construction to the paper's own inputs or self-citations. The assumptions of finite generation and Zariski density are explicitly stated as prerequisites for the algorithms to apply, not outputs. The central contribution is computational and does not rely on load-bearing self-citation chains or ansatzes smuggled from prior work by the same authors. This is a standard non-circular finding for an algorithmic paper in computational group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all details are deferred to the full text which is unavailable here.

pith-pipeline@v0.9.0 · 5637 in / 1026 out tokens · 33447 ms · 2026-05-25T02:35:29.125094+00:00 · methodology

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Reference graph

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