Systoles of Two Extremal Riemann Surfaces

J.R. Quine

The purpose of this note is to study the $\pi_1$ and $H_1$ systoles of two Riemann surfaces, the Klein genus $3$ surface, $y^7=x^2(1-x)$, which we will denote $S_K$, and Bolza's surface of genus $2$, $y^2=x(1-x^4)$, which we will denote $S_B$. Using period matrices, we identify the $H_1$ systoles and show for these examples that the $\pi_1$ and $H_1$ systoles are in one to one correspondence. In their respective Teichm\"uller spaces, $S_B$ is known to be extremal and $S_K$ locally extremal for these systoles.

Key Words and phrases: systole, Riemann surface, Jacobian, period matrix, length spectrum, triangle group, triangle surface, Klein genus 3 curve, symplectic lattice

Classification Numbers Primary: 30F10. Secondary: 11H55