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Random surfaces with large systoles

Mingkun Liu, Bram Petri

Abstract

We present two constructions, both inspired by ideas from graph theory, of sequences random surfaces of growing area, whose systoles grow logarithmically as a function of their area. This also allows us to prove a new lower bound on the maximal systole of a closed orientable hyperbolic surface of a given genus.

Random surfaces with large systoles

Abstract

We present two constructions, both inspired by ideas from graph theory, of sequences random surfaces of growing area, whose systoles grow logarithmically as a function of their area. This also allows us to prove a new lower bound on the maximal systole of a closed orientable hyperbolic surface of a given genus.
Paper Structure (17 sections, 27 theorems, 103 equations, 3 figures)

This paper contains 17 sections, 27 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Random Belyı̆ surfaces
  3. Geometry and topology
  4. Properties of words and their traces
  5. The construction
  6. The proof of Theorem \ref{['thm_linsim_rephrased']}
  7. The maximal systole problem
  8. Counting
  9. A database of surfaces with large systoles
  10. Random regular covers
  11. Non-regular covers
  12. The union bound
  13. Closed base surfaces and groups of Lie type
  14. Non-compact base surfaces and groups of Lie type
  15. Expansion
  16. ...and 2 more sections

Key Result

Theorem 1.1

We have

Figures (3)

  • Figure 1: The systole of random Belyı̆ surfaces with one cusp, generated with a method inspired by ideas of Linial--Simkin, versus the best known upper bound.
  • Figure 2: An example of an initial configuration $X_n^{(0)}$ (with the orientation induced by the page) and its dual graph. The only primitive cycle carries the word $(LR)^n$, which has trace $\left(\frac{3+\sqrt{5}}{2}\right)^n + \left(\frac{3-\sqrt{5}}{2}\right)^n$ and hence $X_n^{(0)}$ can be used as input for Theorem \ref{['thm_linsim_rephrased']}.
  • Figure 3: $A^{[l,m+1]}$: the union of polygons that the segment $\alpha^{[l,m+1]}$ runs through.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3: PW
  • Theorem 2.4
  • Lemma 2.5
  • ...and 42 more