Maximal systole of hyperbolic surface with largest $S^3$ extendable abelian symmetry

TL;DR

This paper derives a formula for the maximal systole length of hyperbolic surfaces with the largest $S^3$-extendable abelian symmetry, involving a complex algebraic expression dependent on genus.

Contribution

It provides an explicit formula for the maximal systole of hyperbolic surfaces with maximal $S^3$-extendable abelian symmetry, a novel result in geometric topology.

Findings

Derived the formula for maximal systole as $2\mathrm{arccosh} K$.

Expressed $K$ explicitly in terms of $L$ and $g$, involving cube roots and square roots.

Connected the systole length to the genus of the surface via $L= 4\cos^2 \frac{\pi}{g-1}$.

Abstract

We give the formula for the maximal systole of the surface admits the largest $S^3$-extendable abelian group symmetry. The result we get is $2\mathrm{arccosh} K$. Here \begin{eqnarray*} K &=& \sqrt[3]{\frac{1}{216}L^3 +\frac{1}{8} L^2 + \frac{5}{8} L - \frac{1}{8} + \sqrt{\frac{1}{108}L(L^2+18L+27)} } & & + \sqrt[3]{\frac{1}{216}L^3 +\frac{1}{8} L^2 + \frac{5}{8} L - \frac{1}{8} - \sqrt{\frac{1}{108}L(L^2+18L+27)} } & & + \frac{L+3}{6}. \end{eqnarray*} and $L= 4\cos^2 \frac{\pi}{g-1}$.