Yue Gao’s research while affiliated with Peking University and other places

We obtain the exact values of the systoles of these hyperbolic surfaces of genus g with cyclic symmetry of the maximum order and the next maximum order. Precisely, for the genus g hyperbolic surface with order 4g + 2 cyclic symmetry, the systole is $\left( {1 + \cos {{\rm{\pi }} \over {2g + 1}} + \cos {{2{\rm{\pi }}} \over {2g + 1}}} \right)$ when g ⩾ 7, and for the genus g hyperbolic surface with order 4g cyclic symmetry, the systole is $\left( {1 + 2\cos {{\rm{\pi }} \over {2g}}} \right)$ when g ⩾ 4.

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 $L= 4\cos^2 \frac{\pi}{g-1}$.

We obtain the exact values of the systoles of these hyperbolic surfaces of genus g with cyclic sysmetries 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 $g\ge4$.