Short homology bases for hyperelliptic hyperbolic surfaces

Given a hyperelliptic hyperbolic surface S of genus g ≥ 2, we find bounds on the lengths of homologically independent loops on S. As a consequence, we show that for any λ ∈ (0, 1) there exists a constant N(λ) such that every such surface has at least \(\left\lceil {\lambda \cdot {2 \over 3}g} \right\rceil \) homologically independent loops of length at most N(λ), 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 \({2 \over 3}g\) linearly independent vectors.