Abstract or Keywords
Given a hyperelliptic hyperbolic surface $S$ of genus $g \geq 2$, we find
bounds on the lengths of homologically independent loops on $S$. As a
consequence, we show that for any $\lambda \in (0,1)$ there exists a constant
$N(\lambda)$ such that every such surface has at least $\lceil \lambda \cdot
\frac{2}{3} g \rceil$ homologically independent loops of length at most
$N(\lambda)$, extending the result in [Mu] and [BPS]. This allows us to extend
the constant upper bound obtained in [Mu] on the minimal length of non-zero
period lattice vectors of hyperelliptic Riemann surfaces to almost $\frac{2}{3}
g$ linearly independent vectors.