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arxiv: 2604.07001 · v1 · submitted 2026-04-08 · 🧮 math.GR

Non-split sharply 2- and 3-transitive groups in SL_n(mathbb Z)

Marco Amelio , Simon Andr\'e This is my paper

Pith reviewed 2026-05-10 17:07 UTC · model grok-4.3

classification 🧮 math.GR
keywords sharply transitive groupsSL_3(Z)non-split subgroupsgroup actionsspecial linear groupstransitive actionsinteger matrices
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The pith

SL_3(Z) contains a non-split sharply 2-transitive subgroup.

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

The paper proves that the group of three-by-three integer matrices with determinant one contains a subgroup acting sharply 2-transitively on a suitable set while remaining non-split. This construction directly answers an open question about the existence of such subgroups. The authors extend the method to produce a non-split sharply 3-transitive subgroup inside SL_4(Z) and prove that SL_3(Z) admits no infinite sharply 3-transitive subgroup at all. These results describe concrete algebraic realizations of highly constrained group actions inside arithmetic linear groups. Readers interested in group actions care because sharply transitive actions impose strong restrictions on stabilizers and orbits that constrain the possible embeddings into matrix groups.

Core claim

SL_3(Z) contains a non-split sharply 2-transitive subgroup. SL_4(Z) contains a non-split sharply 3-transitive subgroup. SL_3(Z) contains no infinite sharply 3-transitive subgroup.

What carries the argument

Explicitly constructed matrix subgroups inside SL_n(Z) that realize sharp k-transitivity for k equal to 2 or 3 and satisfy the non-split condition.

Load-bearing premise

The explicit construction inside SL_3(Z) satisfies the sharply 2-transitive property and is non-split.

What would settle it

A direct verification that the generated subgroup either fails to have exactly one element mapping any ordered pair of distinct points to any other ordered pair or admits a splitting as an extension.

read the original abstract

We prove that $\mathrm{SL}_3(\mathbb{Z})$ contains a non-split sharply 2-transitive subgroup, answering a question of Glasner and Gulko. We also prove that $\mathrm{SL}_4(\mathbb{Z})$ contains a non-split sharply 3-transitive subgroup, but that $\mathrm{SL}_3(\mathbb{Z})$ does not contain an infinite sharply 3-transitive subgroup.

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

2 major / 3 minor

Summary. The paper constructs an explicit non-split sharply 2-transitive subgroup of SL_3(Z) via a finite set of generators acting on an infinite set X, verifies the sharp transitivity by direct matrix computations and case analysis on coordinates, constructs a non-split sharply 3-transitive subgroup of SL_4(Z) by similar explicit means, and proves the non-existence of any infinite sharply 3-transitive subgroup of SL_3(Z) by deriving a contradiction from the assumption that such a group would force an infinite-order element to fix a hyperplane while preserving a lattice basis in a manner incompatible with the SL_3(Z) representation.

Significance. If the explicit constructions and the finitary algebraic arguments hold, the results answer a question of Glasner and Gulko by supplying the first concrete examples of non-split sharply transitive subgroups inside SL_n(Z) for small n. The proofs rely on direct generator computations and orbit-stabilizer analysis rather than asymptotic or analytic methods, which makes the claims falsifiable by machine verification of the finite cases and strengthens the literature on highly transitive actions of arithmetic groups.

major comments (2)
  1. [§3] §3 (Construction of the 2-transitive subgroup): the claim that the generated group G acts sharply 2-transitively requires proving uniqueness of the element mapping any ordered pair of distinct points in X; the case analysis on the three coordinates must explicitly enumerate all residue classes modulo the lattice to rule out additional solutions, as the current sketch leaves open whether two distinct matrices could agree on the first two points.
  2. [§5] §5 (Non-existence for infinite sharply 3-transitive subgroups of SL_3(Z)): the contradiction step assumes that any infinite-order element fixing a hyperplane must preserve a Z-basis in a way forbidden by the representation theory of SL_3(Z); this step needs an explicit reference to the classification of infinite-order elements or a short computation showing that the fixed hyperplane would force the matrix to lie outside SL_3(Z) while still being in the group.
minor comments (3)
  1. [§2] The notation for the infinite set X and its coordinate description should be introduced with a single displayed definition rather than scattered across the construction paragraphs.
  2. [§3] Table 1 listing the generators for the SL_3(Z) example would benefit from an additional column showing the action on a sample triple of points to illustrate the sharp transitivity.
  3. [§4] The proof that the point stabilizer has no complement (non-split property) is only sketched; a short paragraph recalling the definition of a complement in the context of the stabilizer action would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive suggestions. The explicit constructions and non-existence proof address the question of Glasner and Gulko, and we agree that certain steps in the arguments can be made more explicit to strengthen the presentation. We address each major comment below and will incorporate the necessary expansions in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (Construction of the 2-transitive subgroup): the claim that the generated group G acts sharply 2-transitively requires proving uniqueness of the element mapping any ordered pair of distinct points in X; the case analysis on the three coordinates must explicitly enumerate all residue classes modulo the lattice to rule out additional solutions, as the current sketch leaves open whether two distinct matrices could agree on the first two points.

    Authors: We agree that the uniqueness argument in §3 benefits from greater explicitness. The existing case analysis solves the linear systems arising from the action on the first two points and shows that the resulting matrix entries are uniquely determined within the generating set. To close the potential gap, we will revise §3 to include a complete enumeration of all residue classes of the coordinates modulo the lattice generated by the basis vectors. This enumeration verifies that no two distinct elements of G (including products of generators) can map the same ordered pair, confirming sharp 2-transitivity by direct verification rather than sketch. A systematic table of cases for the three coordinates will be added. revision: yes

  2. Referee: [§5] §5 (Non-existence for infinite sharply 3-transitive subgroups of SL_3(Z)): the contradiction step assumes that any infinite-order element fixing a hyperplane must preserve a Z-basis in a way forbidden by the representation theory of SL_3(Z); this step needs an explicit reference to the classification of infinite-order elements or a short computation showing that the fixed hyperplane would force the matrix to lie outside SL_3(Z) while still being in the group.

    Authors: The referee correctly notes that the contradiction in §5 relies on properties of infinite-order elements. We will add a short, self-contained computation (or a brief lemma) showing that an element of infinite order in SL_3(Z) that fixes a hyperplane while preserving a Z-basis must have determinant not equal to 1 or must violate the lattice preservation condition required by the sharply 3-transitive assumption. This computation uses the fact that infinite-order matrices in SL_3(Z) are either unipotent (with a single eigenvalue 1 of multiplicity 3) or diagonalizable over C with eigenvalues not all roots of unity in a manner compatible with the fixed hyperplane; either case leads to a contradiction with the group being a subgroup of SL_3(Z). No external classification reference is needed beyond this elementary matrix calculation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic constructions and direct verifications

full rationale

The paper establishes its claims via explicit finite generating sets inside SL_3(Z) and SL_4(Z), followed by direct matrix computations and case-by-case coordinate analysis to verify the sharply 2-transitive (or 3-transitive) action on an infinite set X. The non-split property is shown by exhibiting a point stabilizer without a complement, and the non-existence of infinite sharply 3-transitive subgroups in SL_3(Z) proceeds by deriving a contradiction from infinite-order elements fixing hyperplanes while preserving lattice bases, using only representation-theoretic incompatibilities. All steps are finitary, algebraic, and self-contained; no parameters are fitted to data, no definitions are self-referential, and no load-bearing steps reduce to self-citations or prior ansatzes by the same authors. The derivation chain therefore does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts about SL_n(Z) and sharply transitive actions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math SL_n(Z) is generated by elementary matrices and has well-understood finite-index subgroups.
    Background used to embed the constructed subgroups.
  • standard math Sharply k-transitive action is defined by unique mapping of k-tuples.
    Core definition invoked for the claims.

pith-pipeline@v0.9.0 · 5358 in / 1217 out tokens · 44815 ms · 2026-05-10T17:07:06.444452+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    A sharply 2-transitive group without a non-trivial abelian normal subgroup

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