Filling surfaces with very few systoles

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.

• The paper demonstrates that for infinitely many genera, hyperbolic surfaces can be filled by their systoles with at most approximately 11.85 g/ln g elements.
• It employs an innovative construction using congruence covers of a genus 2 surface and arithmetic Fuchsian groups to derive explicit systole bounds.
• The approach leverages geometric decompositions, Penner systems, and tessellations to tightly close the gap between previous upper and lower bounds.

Filling Surfaces with Very Few Systoles: A Sharp Bound for Systole-Filling Sets

Introduction and Motivation

The study of systoles—shortest non-contractible closed geodesics—on hyperbolic surfaces encapsulates both geometric and topological properties of surfaces of genus $g \geq 2$. A set of geodesics is said to fill a surface if their complement is a union of polygons. Thurston established that there exist hyperbolic surfaces whose systoles themselves fill, but quantifying the minimal size of such filling sets—denoted $Fill(g)$—has been the focus of significant research efforts.

Earlier works provided polynomial and subpolynomial upper bounds on $Fill(g)$, yet a notable gap persisted from the Anderson, Parlier, and Pittet lower bound of asymptotic order $\pi\, g / \ln g$ [APP]. This paper by Olivier Mathieu provides a construction matching the lower bound up to explicit constants, thereby resolving this gap and establishing tight asymptotics.

Main Results

The central theorem asserts that for infinitely many genera $g$, there exist closed orientable hyperbolic surfaces of genus $g$ filled by their systoles with cardinality at most

$$Fill(g) < 9\ln(2 + \sqrt{3})\, \frac{g}{\ln g} \approx 11.85\, \frac{g}{\ln g}$$

This matches the form of the Anderson-Parlier-Pittet lower bound, and the explicit constant is deduced through arithmetic and geometric arguments involving congruence covers of certain genus 2 surfaces. The construction applies to an infinite family of genera, thus showing that the lower bound is sharp in order.

The approach substantially improves upon previous upper bounds, which were at best of order $o\left(\frac{g}{\sqrt{\ln g}}\right)$ for infinitely many $g$ [FB20, IM], and provides a remarkably transparent construction.

Construction Overview

The construction is rooted in the exploitation of arithmetic properties of a particular genus 2 surface $S$ that admits a tessellation by four regular right-angled hexagons. The systoles of this surface are explicitly described and realized as the one-skeleton of the tessellation, each of length $Fill(g)$0.

To generate large genus surfaces with controllably few systoles, congruence covers $Fill(g)$1 of $Fill(g)$2 are considered. These are associated to congruence subgroups $Fill(g)$3 determined by arithmetic in a quaternion algebra over $Fill(g)$4, with explicit use of sequences derived from powers of $Fill(g)$5.

A critical lemma establishes that the systoles of $Fill(g)$6 are precisely the connected components of preimages, under the covering map, of the systoles of $Fill(g)$7, and their length grows linearly with $Fill(g)$8. The number of systoles in $Fill(g)$9 is thus explicitly computable, yielding the bound

$Fill(g)$0

for genus $Fill(g)$1.

The constant is refined further by leveraging minimal filling subsets constructed in the degree 16 covering $Fill(g)$2 (of genus 17): a recent result [AII] demonstrates the existence of filling subsets with only 24 elements among its 48 systoles, allowing replacement of the constant 18 by 9. The transfer of these minimal filling sets to higher covers through the covering correspondence yields the desired bound.

Theoretical Implications

This result closes a longstanding asymptotic question on the minimal number of systoles needed to fill hyperbolic surfaces of large genus, providing both existential and explicit constructions for infinitely many genera.

It demonstrates the power of arithmetic and combinatorial techniques—particularly congruence coverings and careful accounting of systole preimages—in answering delicate extremal questions in the geometry of Riemann surfaces. The approach also exploits intersection patterns represented via "Penner systems" and the colorability of systoles, providing a direct link between geometric decompositions and algebraic group data.

Numerical and Structural Highlights

• The explicit upper bound constant is $Fill(g)$3, matching the lower bound in order.
• The construction provides examples for infinitely many genera, including explicit genus computations for lower genera (e.g., $Fill(g)$4).
• The systole lengths scale with the degree of the cover, imparting both length and cardinality control through arithmetic progression.
• The construction employs arithmetic Fuchsian groups realized via a quaternion algebra over $Fill(g)$5 and explores their congruence subgroups.

Future Directions

The result prompts further investigations into the distribution, interaction, and generation properties of systole sets on arithmetic and non-arithmetic surfaces. It raises the question whether similar sharp asymptotic results hold for all sufficiently large genera or for more restrictive families of surfaces, such as non-arithmetic ones.

On the theoretical side, the methods suggest applications in constructing surfaces with prescribed geometric decompositions and in the study of automorphism group actions on moduli space. There may also be implications for algorithmic recognition of systole-filling sets and explicit constructions of surfaces with extremal geometric properties.

Refinements could involve minimizing the constant further or achieving sharp results for all genera, as well as extending techniques to higher-dimensional analogues or nonhyperbolic metrics.

Conclusion

This paper provides a precise asymptotic determination of the minimal number of systoles required to fill hyperbolic surfaces of genus $Fill(g)$6, matching known lower bounds up to an explicit constant, via an arithmetically grounded and geometrically transparent construction. The work cements the systole-filling problem as fundamentally arithmetic in nature and establishes a pathway for further advances in the study of extremal geometry on Riemann surfaces.

Reference: "Filling surfaces with very few systoles" (2606.28954)