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arxiv: 2606.28954 · v1 · pith:FSVMSINBnew · submitted 2026-06-27 · 🧮 math.MG · math.DG

Filling surfaces with very few systoles

Olivier Mathieu This is my paper

Pith reviewed 2026-06-30 08:23 UTC · model grok-4.3

classification 🧮 math.MG math.DG
keywords hyperbolic surfacessystolesgenusfilling setsclosed geodesicsasymptotic boundsmetric geometry
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The pith

Hyperbolic surfaces of genus g can be filled by O(g / ln g) systoles, matching the Anderson-Parlier-Pittet lower bound.

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

The paper constructs hyperbolic surfaces where the systoles fill the surface and their total number is bounded by O(g / ln g). This achieves the lower bound previously shown by Anderson, Parlier and Pittet while improving on earlier upper bounds of o(g / sqrt(ln g)). The construction is described as simpler than those in prior work. A reader would care because it establishes that the minimal number of shortest closed geodesics sufficient to fill the surface is asymptotically tight at order g / ln g.

Core claim

The paper describes hyperbolic surfaces filled by their systoles where the total number of systoles is in O(g / ln g), which is equivalent to the lower bound of Anderson, Parlier and Pittet. Previous upper bounds were in o(g / sqrt(ln g)). The present approach is simpler than the methods of earlier papers.

What carries the argument

A construction of hyperbolic surfaces whose systoles fill the surface while keeping their number in O(g / ln g).

If this is right

  • The minimal number of systoles needed to fill a hyperbolic surface of genus g is Theta(g / ln g).
  • Simpler constructions suffice to reach the optimal asymptotic count for systole-filling surfaces.
  • The gap between upper and lower bounds on filling systoles is now closed up to constants.

Where Pith is reading between the lines

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

  • Similar counting arguments might apply to filling sets in other geometries such as Euclidean or spherical surfaces.
  • One could check whether random hyperbolic surfaces of large genus achieve filling with comparably few systoles.
  • The result suggests examining the trade-off between systole count and filling efficiency in variable-curvature metrics.

Load-bearing premise

A construction exists that realizes the O(g / ln g) count while ensuring the systoles actually fill the surface without post-hoc adjustments or unstated constraints.

What would settle it

An explicit example of a hyperbolic surface of genus g whose filling systoles number o(g / ln g), or a verification that the constructed surfaces fail to fill under the stated metric.

read the original abstract

In the paper we describe hyperbolic surfaces filled by their systoles, where the total number of systoles is in $O(\frac{g}{\ln \,g})$, that is equivalent to the lower bound of Anderson, Parlier and Pittet \cite{APP}. Various papers \cite{SS}\cite{FB20}\cite{Sanki}\cite{ IM}\cite{ Mathieu} have investigated the same question, and the best previously known upper bounds where in $o(\frac{g}{{\sqrt{\ln \,g}}})$. Surprizingly the present approach is, in our opinion, much simpler than the methods of earlier papers.

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

1 major / 0 minor

Summary. The manuscript claims to construct hyperbolic surfaces of genus g filled by their systoles with the total number of systoles in O(g / ln g), matching the lower bound of Anderson, Parlier and Pittet. It states that prior upper bounds were o(g / sqrt(ln g)) and describes the new approach as simpler than those in the cited works.

Significance. An explicit construction achieving the optimal asymptotic order for the number of filling systoles would close the gap to the known lower bound and simplify prior techniques in systolic geometry.

major comments (1)
  1. [Abstract] Abstract: the central claim is the existence of a construction realizing O(g / ln g) filling systoles, but the manuscript supplies no metric, topology, systole list, or verification that the systoles fill the surface. Without these elements the asserted bound and filling property cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment concerns the absence of explicit construction details in the manuscript. We address this below and agree that revisions are needed to make the claims verifiable.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is the existence of a construction realizing O(g / ln g) filling systoles, but the manuscript supplies no metric, topology, systole list, or verification that the systoles fill the surface. Without these elements the asserted bound and filling property cannot be checked.

    Authors: We agree that the current manuscript text does not supply an explicit metric, a concrete topology (such as a pants decomposition or fundamental domain), a list of systole curves, or a direct verification that these curves fill the surface. The abstract and introductory paragraph assert the existence of such a construction achieving the Anderson-Parlier-Pittet bound, but without the supporting details the claim cannot be checked. In the revised version we will add a self-contained description of the metric (via explicit hyperbolic gluings), the topology for a sequence of genera, the systole curves, and a proof that their union is filling. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract presents a construction of hyperbolic surfaces whose systoles fill the surface and achieve the count O(g / ln g), stated as matching the external lower bound from Anderson, Parlier and Pittet. Prior works (including one self-citation) are referenced only for historical upper bounds; the new result is framed as an explicit construction rather than a derivation that reduces to fitted parameters, self-definitions, or load-bearing self-citations. No equations or steps are supplied that would allow reduction of the claimed count or filling property to the inputs by construction. The result is therefore treated as self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no specific free parameters, axioms beyond standard background, or invented entities are identifiable.

axioms (1)
  • standard math Standard axioms and definitions of hyperbolic geometry and systoles
    Background for the existence of hyperbolic surfaces and systoles.

pith-pipeline@v0.9.1-grok · 5618 in / 1122 out tokens · 35120 ms · 2026-06-30T08:23:35.412654+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 2 canonical work pages · 1 internal anchor

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