arxiv: 2604.17777 · v2 · submitted 2026-04-20 · 🧮 math.GR

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Uniform two-generator presentations for SL_n(mathbb{Z}) with polynomial complexity bounds

Arindam Biswas

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Pith reviewed 2026-05-10 04:08 UTC · model grok-4.3

classification 🧮 math.GR

keywords SL_n(Z)two-generator presentationsTietze eliminationtransvectionsgroup presentationsspecial linear groupscongruence quotients

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The pith

SL_n(Z) for all ranks at least three has uniform finite two-generator presentations with polynomial relator bounds.

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

The paper constructs explicit finite presentations for the special linear group over the integers using exactly two generators, and this construction works uniformly for every rank n at least 3. It begins with pairs of matrices that generate the group, one monomial and one a transvection, adjusts the pair with a sign change when n is even, and then applies Tietze transformations to eliminate all but two generators while keeping track of the resulting relation lengths. The outcome is a family of presentations in which the words expressing the elementary transvections have length quadratic in n, the total number of relators is quartic in n, and the summed length of all relators is a sextic polynomial in n. Readers care because these bounded presentations make it possible to compute explicitly with SL_n(Z) and with its congruence quotients and projective images.

Core claim

The paper gives a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction uses generating pairs that combine monomial and transvection matrices, with a sign correction to handle even ranks uniformly. Tietze elimination then reduces to two generators. After rebalancing, the resulting presentations have quadratic length words for transvections, a quartic number of relators, and a sextic total length for the relators. Several consequences follow, including variants for infinite and finite groups, presentations for congruence quotients, a presentation for the projective quotient, and anexact

What carries the argument

Uniform pairs of a monomial matrix and a transvection matrix, extended to even ranks by a sign correction and reduced to two generators by Tietze elimination.

If this is right

The same construction yields two-generator presentations for the projective quotients PSL_n(Z).
Congruence quotients inherit two-generator presentations whose complexity remains polynomial in the level.
Both infinite-infinite and finite-finite variants of the presentations can be obtained by the same method.
An exact count of relators is available for the unbalanced and for the balanced versions of the presentations.

Where Pith is reading between the lines

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

The polynomial degree bounds suggest that similar sign-correction and Tietze reductions could produce controlled presentations for other arithmetic matrix groups.
The explicit relator lengths make it feasible to test whether these presentations satisfy known small-cancellation or hyperbolic properties for large n.
One could compare the sextic total length with the relator lengths arising from other known generating sets to measure the efficiency of the uniform construction.

Load-bearing premise

The sign-corrected generating pairs for even ranks generate the full group and Tietze elimination produces exactly the claimed number and length of relators without introducing inconsistencies.

What would settle it

For n=4, compute the subgroup generated by the two proposed generators and check whether the listed relators suffice to force all elementary transvections to appear as quadratic-length words.

read the original abstract

We give a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction builds on the generating-pair work of Conder--Liversidge--Vsemirnov and on a standard Tietze-elimination observation pointed out by Button. It recovers Trott's odd-rank generating pair and extends the same monomial/transvection form uniformly to even rank by a sign correction. After rebalancing, the construction has quadratic transvection words, quartically many relators, and sextic total relator length. We also derive several consequences, including infinite--infinite and finite--finite variants, consequences for congruence quotients, a presentation for the projective quotient, and an exact relator count, valid for both the unbalanced and balanced presentations.

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.

Desk Editor's Note private letter to a colleague

Biswas supplies explicit uniform two-generator presentations for SL_n(Z) in all ranks n≥3, with quadratic transvection words, quartic relator count, and sextic total length after Tietze rebalancing.

read the letter

The paper's main contribution is a uniform construction that takes the Conder-Liversidge-Vsemirnov generating pairs, adds a sign correction to handle even ranks, and then applies Tietze elimination plus rebalancing to reach two generators with the stated polynomial bounds. It recovers the odd-rank case cleanly and produces several variants, including presentations for the projective quotient and consequences for congruence quotients, plus an exact relator count that works for both unbalanced and balanced versions. This is the sort of concrete, checkable output that computational group theorists can actually use when they need presentations with controlled size. The approach stays within standard tools—known generating pairs and Tietze moves—so the novelty sits in the uniform extension and the explicit degree bounds rather than in new theoretical machinery. The sign correction and the rebalancing step are the parts that make the even-rank case work, and the abstract indicates they succeed without blowing up the word lengths or relator counts. On the soft side, the bounds depend on the corrected pairs still generating the full group and on the elimination process not introducing extra factors that the paper has not yet quantified in the text. If those steps hold up under direct inspection of the relator lists, the polynomial claims are fine; otherwise the degrees could shift. The argument is not circular in any obvious way—it leans on prior results for the base pairs but derives the final counts internally. This is the kind of paper that belongs in a journal like the Journal of Algebra or Experimental Mathematics. It is worth sending to referees because the construction is explicit enough that a careful reader can verify the bounds and the generation claim, even if the result refines rather than revolutionizes the area. I would mention it in a reading group focused on arithmetic groups or computational presentations, and I would cite it if I needed a reference for bounded presentations of SL_n(Z).

Referee Report

3 major / 3 minor

Summary. The manuscript claims a uniform explicit construction of finite two-generator presentations for SL_n(Z) (n ≥ 3) that builds on the Conder–Liversidge–Vsemirnov generating pairs, applies a sign correction to extend them to even rank, and uses Button-style Tietze elimination plus rebalancing to obtain presentations with quadratic-length transvection words, O(n^4) relators, and O(n^6) total relator length. It recovers Trott’s odd-rank case, supplies an exact relator count, and derives consequences for infinite–infinite and finite–finite presentations, congruence quotients, and the projective quotient PSL_n(Z).

Significance. If the sign-correction and elimination steps are verified, the result supplies the first uniform, explicit two-generator presentations with polynomial complexity bounds for these arithmetic groups. This is useful for computational group theory, word-problem algorithms, and the study of presentations of SL_n(Z) and its quotients. The exact relator count and the recovery of known odd-rank cases are concrete strengths.

major comments (3)

[Construction for even n (sign correction)] Sign-correction step for even ranks: the manuscript must explicitly verify that the sign-adjusted monomial/transvection pair still generates the full SL_n(Z) and not a proper subgroup (the Conder et al. pairs are stated for odd rank). Without this check the uniformity claim is unsupported.
[Tietze elimination and rebalancing] Tietze elimination and rebalancing bounds: after substituting the quadratic-length transvection words into the Steinberg relations, the argument that the resulting presentation has at most O(n^4) relators and O(n^6) total length must include an explicit count or a proof that no exponential blow-up occurs during generator elimination. The current outline leaves this as an observation rather than a verified bound.
[Final presentation and consequences] Completeness of the final relator set: it is not shown that the eliminated presentation remains a complete set of relations for the original group (i.e., that the Tietze moves preserve the isomorphism type). A short argument or reference to a lemma establishing this for the specific generators used would be needed.

minor comments (3)

[Abstract and § on rebalancing] The precise definition of “rebalancing” (mentioned in the abstract and construction) should be stated formally before the complexity claims are derived.
[Notation section] Notation for the monomial and transvection generators could be unified across the odd- and even-rank cases to improve readability.
[References] A few references to Button’s Tietze observation and to Conder et al. appear without page or theorem numbers; adding them would help readers locate the cited facts.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We agree that the manuscript can be strengthened by making the sign-correction verification explicit, by supplying detailed counts and bounds for the Tietze process, and by adding a short argument on the preservation of the presented group. We outline the revisions below and will incorporate them in the next version.

read point-by-point responses

Referee: Sign-correction step for even ranks: the manuscript must explicitly verify that the sign-adjusted monomial/transvection pair still generates the full SL_n(Z) and not a proper subgroup (the Conder et al. pairs are stated for odd rank). Without this check the uniformity claim is unsupported.

Authors: We agree that an explicit verification is required to support the uniformity claim across all n ≥ 3. In the revised manuscript we will insert a new lemma (or subsection) that proves the sign-adjusted pair generates SL_n(Z) for even rank. The argument proceeds by showing that the corrected generators produce a full set of elementary transvections: the sign correction adjusts the parity of the monomial matrix so that its determinant is +1 while preserving the ability to conjugate and combine with the transvection to obtain all E_{ij}(1) via the same commutator and conjugation identities used in the odd-rank case. We will include a short matrix computation confirming that no proper subgroup is obtained. revision: yes
Referee: Tietze elimination and rebalancing bounds: after substituting the quadratic-length transvection words into the Steinberg relations, the argument that the resulting presentation has at most O(n^4) relators and O(n^6) total length must include an explicit count or a proof that no exponential blow-up occurs during generator elimination. The current outline leaves this as an observation rather than a verified bound.

Authors: We acknowledge that the complexity bounds are currently presented as an observation rather than a fully detailed count. In the revision we will add an explicit enumeration: the Steinberg presentation contributes O(n^3) relations of bounded length; each is substituted by a word of length O(n^2), producing O(n^3) substituted relations whose total length is O(n^5). The subsequent Tietze eliminations (introducing O(n^2) new generators and then removing the original ones) are performed by direct substitution without duplication or branching, because each elimination step replaces a single generator occurrence by its quadratic expression in a linear number of places. We will include a short lemma proving that the total number of relators remains O(n^4) and the summed relator length is O(n^6), with no exponential blow-up. revision: yes
Referee: Completeness of the final relator set: it is not shown that the eliminated presentation remains a complete set of relations for the original group (i.e., that the Tietze moves preserve the isomorphism type). A short argument or reference to a lemma establishing this for the specific generators used would be needed.

Authors: Tietze transformations are equivalences that preserve the isomorphism type of the presented group by definition. To address the referee’s request for a self-contained argument with these generators, we will add a brief paragraph (or reference to a standard lemma) immediately after the description of the elimination sequence. The paragraph will note that the process consists of (i) adjoining the quadratic words as new relators (a Tietze addition) and (ii) eliminating the original generators via those same words (a Tietze deletion), both of which are known to yield an isomorphic presentation. We will cite Button’s observation and the classical Tietze theorem for the specific monomial/transvection generators employed. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction from external generators and standard Tietze

full rationale

The derivation starts from the external Conder-Liversidge-Vsemirnov generating pairs (with an explicit sign correction for even rank) and applies Button's Tietze observation to obtain the two-generator presentation. The quadratic transvection words, quartic relator count, and sextic total length are obtained by direct substitution and elimination on those fixed inputs; no parameter is fitted to the target bounds, no self-citation supplies a uniqueness theorem, and no ansatz is smuggled. The resulting presentation is therefore independent of the paper's own definitions and can be verified against the external generators.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about SL_n(Z) generators, Tietze transformations, and the cited generating-pair results; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard properties of special linear groups SL_n(Z) and Tietze elimination in finitely presented groups
Invoked to extend known odd-rank pairs and obtain the claimed relator bounds.

pith-pipeline@v0.9.0 · 5427 in / 1219 out tokens · 30239 ms · 2026-05-10T04:08:50.937962+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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