Short homology bases for hyperelliptic hyperbolic surfaces

TL;DR

This paper establishes bounds on the lengths of homologically independent loops on hyperelliptic hyperbolic surfaces, showing that a linear proportion of such loops can be uniformly bounded in length, extending previous results.

Contribution

It provides new bounds on homologically independent loops of bounded length on hyperelliptic hyperbolic surfaces, extending prior work to a larger proportion of such loops.

Findings

At least rac{rac{2}{3}g}{}$ homologically independent loops of bounded length exist.

Extended bounds on minimal length of period lattice vectors to rac{rac{2}{3}g}{}$ independent vectors.

Established linear bounds on loop lengths proportional to genus g.

Abstract

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.