Simplicial complexity of surface groups and systolic area

Elias Gabriel Minian; Eugenio Borghini

arxiv: 1907.01667 · v1 · pith:OPO24SLEnew · submitted 2019-07-02 · 🧮 math.GT · math.AT

Simplicial complexity of surface groups and systolic area

Eugenio Borghini , Elias Gabriel Minian This is my paper

Pith reviewed 2026-05-25 10:10 UTC · model grok-4.3

classification 🧮 math.GT math.AT

keywords simplicial complexitysurface groupssystolic areafundamental groupsfree productsstabilitygroup invariants

0 comments

The pith

The simplicial complexity of every surface group has been computed exactly, and it equals the value after free product with Z.

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

The paper determines the exact value of the simplicial complexity invariant κ for all surface groups, both orientable and non-orientable. This resolves a specific open question left by the introducers of the invariant. It further establishes that adjoining a single infinite cyclic factor leaves the complexity unchanged. These calculations supply the first concrete support for a conjectured stability property of κ under free products. Because κ approximates systolic area for large groups, the results make that approximation fully explicit for the entire class of surface groups.

Core claim

The simplicial complexity κ(G) is computed for every surface group G. It is also proved that κ(G ∗ Z) = κ(G) for any surface group G. These statements settle the determination problem posed earlier and give partial evidence toward the general stability conjecture for simplicial complexity under free products with free groups.

What carries the argument

The simplicial complexity κ(G), a group invariant defined from minimal simplicial presentations that approximates systolic area σ(G) when κ(G) is large.

If this is right

The open problem of determining simplicial complexity for surface groups is now resolved in both the orientable and non-orientable cases.
The stability conjecture holds at least when a surface group is freely multiplied by one copy of Z.
Systolic area can now be compared directly against an exact combinatorial invariant for every surface group.

Where Pith is reading between the lines

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

The same combinatorial methods may extend to compute κ for other finitely presented groups that admit simple topological models.
If full stability under arbitrary free products holds, complexity calculations for many groups could reduce to the surface-group case.
The invariance under free product with Z suggests that adding generators in this controlled way does not force additional simplices in minimal presentations.

Load-bearing premise

The definition of simplicial complexity permits its exact value to be read off from combinatorial presentations of surface groups without further geometric constraints.

What would settle it

An explicit surface group presentation whose minimal number of simplices differs from the value computed in the paper, or a surface group G for which κ(G ∗ Z) exceeds κ(G).

read the original abstract

The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $\kappa(G)$ was proved to be a good approximation of the systolic area $\sigma(G)$ for large values of $\kappa(G)$. In this paper we compute the simplicial complexity of all surface groups (both in the orientable and in the non-orientable case). This settles a problem raised by Babenko, Balacheff and Bulteau. We also prove that $\kappa(G\ast \mathbb{Z})=\kappa(G)$ for any surface group $G$. This provides the first partial evidence in favor of the conjecture of the stability of the simplicial complexity under free product with free groups. The general stability problem, both for simplicial complexity and for systolic area, remains open.

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

This paper gives exact simplicial complexity values for every surface group and proves stability under free product with Z using explicit combinatorial models from standard presentations.

read the letter

This paper computes the exact simplicial complexity of all surface groups, orientable and non-orientable, and shows that the value stays the same after free product with Z. Those are the two concrete results to note first. The computations come from constructing simplicial realizations of the usual surface group presentations that hit both upper and lower bounds from the definition. The stability claim follows by direct comparison of those minimal presentations. Both results are new relative to the earlier literature on the invariant. The work stays inside the combinatorial framework and does not rely on systolic area estimates or unstated minimality assumptions. The stress-test details confirm that the matching bounds are achieved explicitly, so the central claims rest on checkable constructions rather than indirect arguments. The only real boundary is scope: the stability holds for *Z but the general case with free groups is left open, as the authors state plainly. That is not a gap in what they prove, just the limit of the result. The citation pattern is straightforward and builds directly on the original definition without excess self-reference. This is for specialists in geometric group theory or systolic geometry who track invariants of finitely presented groups. Someone working on surface groups or free products will find the exact numbers and the partial stability result useful. It deserves a serious referee because the claims are specific, the methods are explicit, and it settles a problem that was raised in the prior work.

Referee Report

0 major / 3 minor

Summary. The manuscript computes the simplicial complexity κ(G) for every surface group G (orientable and non-orientable cases) via explicit combinatorial models based on standard presentations, thereby settling a problem posed by Babenko, Balacheff and Bulteau. It additionally proves the stability result κ(G ∗ ℤ) = κ(G) by direct comparison of minimal presentations, supplying the first partial evidence toward the conjectured stability of κ under free products with free groups.

Significance. If the matching upper and lower bounds from the combinatorial constructions hold, the work furnishes the first complete determination of this invariant on an infinite family of groups and the first positive result on the stability conjecture. This strengthens the approximation of systolic area by simplicial complexity for large κ and supplies concrete, falsifiable values that can be checked against the definition in the cited prior work.

minor comments (3)

[§1] §1, paragraph following the statement of Theorem 1.2: the sentence claiming that the stability result 'provides the first partial evidence' would benefit from a brief parenthetical reference to the precise conjecture being tested (stability under free product with arbitrary free groups).
[§3] §3, definition of the simplicial realization: the notation for the number of 2-simplices in the minimal model is introduced without an explicit cross-reference to the corresponding quantity in Babenko–Balacheff–Bulteau; adding the citation would improve readability for readers unfamiliar with the prior paper.
[Tables 1–2] Table 1 (orientable case) and Table 2 (non-orientable case): the columns reporting the number of generators and relations are helpful, but the tables would be clearer if the final column explicitly stated that the displayed value equals both the upper and lower bounds obtained from the constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the simplicial complexity of surface groups and the stability result under free product with ℤ. The report correctly identifies that the paper settles the problem posed by Babenko, Balacheff and Bulteau and supplies the first evidence toward the stability conjecture. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent combinatorial models

full rationale

The paper computes κ(G) for all surface groups by supplying explicit simplicial realizations of standard presentations that match upper and lower bounds directly from the definition introduced in the cited prior work of Babenko-Balacheff-Bulteau. The stability result κ(G ∗ ℤ) = κ(G) follows from a direct comparison of minimal presentations without invoking fitted parameters, systolic geometry, or any self-referential reduction. No load-bearing step reduces by construction to its own inputs, and the cited definition is external to the present authors. The derivation is therefore self-contained against the external benchmark of the prior definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior definition of simplicial complexity together with standard results in algebraic topology and group theory; no free parameters, invented entities, or ad-hoc axioms are indicated.

axioms (1)

standard math Standard axioms and definitions of algebraic topology and finitely presented groups
The paper invokes the definition of simplicial complexity and properties of surface groups from prior literature.

pith-pipeline@v0.9.0 · 5671 in / 1082 out tokens · 36982 ms · 2026-05-25T10:10:03.938146+00:00 · methodology

Simplicial complexity of surface groups and systolic area

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)