Systoles of hyperbolic surfaces with big cyclic symmetry

TL;DR

This paper precisely calculates the shortest non-contractible loops (systoles) on hyperbolic surfaces with maximal and near-maximal cyclic symmetries, providing explicit formulas for different genera.

Contribution

It derives exact formulas for systoles of hyperbolic surfaces with specific cyclic symmetries, advancing understanding of geometric properties of symmetric hyperbolic surfaces.

Findings

Exact systole values for genus g surfaces with order 4g+2 symmetry

Explicit systole formulas for genus g surfaces with order 4g symmetry

Systole formulas valid for g ≥ 7 and g ≥ 4 respectively

Abstract

We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+\cos \frac{\pi}{2g+1}+\cos \frac{2\pi}{2g+1})$ when $g\ge 7$, and for genus $g$ hyperbolic surface with order $4g$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+2\cos \frac{\pi}{2g})$ when $g\ge4$.