Expected covering radius of a translation surface (1809.10769v3)

Abstract: A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface in any stratum H_1(kappa). The covering radius of a translation surface is the largest radius of an immersed disk. In the case of the stratum H_1(2g-2) of translation surfaces of genus g with one singularity, the covering radius is comparable to the diameter. We show that the expected covering radius of a surface is bounded above by a uniform multiple of ((log g)/g)^1/2, independent of the stratum. This is smaller than what one would expect by analogy from the result of Mirzakhani about the expected diameter of a hyperbolic metric on a Riemann surface. To prove our result, we need an estimate for the volume of the thin part of H_1(kappa) which is given in the appendix.

• The paper derives an upper bound on the expected covering radius of translation surfaces, demonstrating it scales as √(log g/g) for surfaces in a designated stratum.
• It employs a blend of analytic, combinatorial, and measure-theoretic methods, including volume calculations and the Siegel–Veech formalism.
• The findings challenge classical hyperbolic analogies and set the stage for further research into geometric statistics and singularity distributions in moduli spaces.

An Examination of the Expected Covering Radius of Translation Surfaces

This paper presents an in-depth investigation into the geometric properties of translation surfaces, specifically focusing on their expected covering radius. A translation structure on a Riemann surface introduces a singular flat metric, which simplifies the paper of certain geometric properties like the covering radius, defined as the largest radius of an immersed disk that can fit on the surface without encountering any singularities.

Main Results and Methods

The authors aim to establish an upper bound on the expected value of the covering radius of a translation surface within a specified stratum. They demonstrate that for the stratum $M_1(\kappa)$, the upper bound is proportional to $\sqrt{\log g / g}$, where $g$ is the genus of the surface. Intriguingly, this bound is more stringent than expected when compared with analogies drawn from Mirzakhani's results concerning the hyperbolic genus concerning the expected diameter.

The proofs involve a blend of analytic and combinatorial methods, including volume calculations of the moduli space's thin parts and combinatorial counting principles concerning the surface's genus and singularity types. The authors dissect translation surfaces into four distinct cases—based on disk embeddings and cylindrical structures—to tackle the measure of the expected covering radius across these configurations. They use elegant measure-theoretic arguments to handle the case of large embedded disks and apply the Siegel--Veech formalism to estimate contributions from cylinders and saddle connections.

Numerical Analysis and Theorems

Numerical estimates are pivotal in the authors' arguments, leveraging known bounds on the volume of moduli spaces and recursive formulas for translation surfaces' volumes. The statements about large genus behavior, particularly in contrast with expectations from hyperbolic geometry, provide a nuanced view of geometric statistics on moduli spaces.

For instance, when focusing on strata with singularity configurations such as $M_1(2g-2)$ (surfaces of genus $g$ with one singularity), the authors derive an upper bound for the expected diameter, relying on comparing it with the uniform distribution on the moduli space.

Implications and Future Directions

The paper's results highlight that the classical views of hyperbolic metric properties do not always straightforwardly apply to translation surfaces. This discrepancy invites further research into understanding the deeper geometric structures and statistical characteristics of moduli spaces. A more profound understanding of these properties could inspire novel statistical tools in the analysis of geometric structures, as well as improved computational methods within the field of modern algebraic geometry and Teichmüller theory.

The methods and results also suggest potential extensions to other geometric settings and singularity structures, prompting questions about the universality of these findings. Furthermore, the connection between covering radius, moduli space volume, and singularity distribution could be further explored to derive tighter bounds and broader applicability across other fields of mathematics, including number theory and algebraic geometry focuses.

Conclusion

In summarizing this detailed inquiry, the authors shed light on under-explored aspects of translation surface geometry, providing robust estimates and a framework for future explorations. Their approach via Siegel--Veech constants, combined with geometric volume estimates, contributes valuable depth to the understanding of translation surfaces' geometric statistics and opens pathways for extensive research and theoretical development in related fields.