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# Short homology bases for hyperelliptic hyperbolic surfaces

Jun 2022

Given a hyperelliptic hyperbolic surface $S$ of genus $g \geq 2$, we findbounds on the lengths of homologically independent loops on $S$. As aconsequence, 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 \cite{mu1} and \cite{bps}. This allows usto extend the constant upper bound obtained in \cite{mu1} on the minimal lengthof non-zero period lattice vectors of hyperelliptic Riemann surfaces to almost$\frac{2}{3} g$ linearly independent vectors.

Q1论文试图解决什么问题？
Q2这是否是一个新的问题？
Q3这篇文章要验证一个什么科学假设？
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