Short homology bases for hyperelliptic hyperbolic surfaces

Abstract

Given a hyperelliptic hyperbolic surface $S$ of genus $g\ge 2$, 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.

19 pages, 8 figures, proof of Lemma 3.2 corrected