Connectedness of quadratic hyperelliptic square-tiled surfaces via the paper’s methods

Show the connectedness of $ST^{hyp}_{quad}(\mu)$ under cylinder shears by extending the paper’s reconfiguration techniques to quadratic hyperelliptic components.

Background

The paper develops a quotient-by-involution approach for Abelian hyperelliptic components and uses spherical reconfiguration results to prove their connectedness. Quadratic hyperelliptic components have a different quotient structure, with two distinguished singularities rather than one, and the same reduction is not directly available.

The remark records an explicit unresolved limitation: the authors were unable to use their techniques to prove connectedness in this quadratic setting.

References

However, we were not able to apply the techniques we develop in this article in order to show the connectedness of $ST{hyp}_{quad}(\mu)$.

— Reconfiguration of square-tiled surfaces  (2501.15978 - Delecroix et al., 27 Jan 2025) in Remark 6.1, Section “Hyperelliptic components”