Systoles of Two Extremal Riemann Surfaces<P>

J.R. Quine<P>

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.<P>

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

Classification Numbers Primary: 30F10. Secondary:  11H55<P>