Connectedness of quadratic hyperelliptic components

Determine whether the quadratic hyperelliptic components of square-tiled surfaces are connected under cylinder shears.

Background

Quadratic hyperelliptic components correspond to quadratic square-tiled surfaces whose quotient by the hyperelliptic involution has a spherical profile with two distinguished singularities of sufficiently high degree. The paper’s reduction methods establish connectedness for Abelian hyperelliptic components, but the analogous quadratic case is not covered.

The authors specifically identify connectedness under ordinary cylinder shears—not merely half-cylinder shears—as an unresolved question. Their method for reaching path-like configurations requires restrictions on the profile that are not satisfied by quadratic hyperelliptic components.

References

The first open problem we consider is naturally \cref{conj:squareTiledSurfacesConnectedComponents} in full generality. In this article, we confirmed \cref{conj:squareTiledSurfacesConnectedComponents} in two special cases. First, in the Abelian hyperelliptic components (see \cref{thm:connectednessHyperelliptic}). Second, on the sphere (see \cref{thm:connectednessSphere}), in all strata when authorizing half-shears, but only in the strata such that $\mu_1 \le 1$ and $(\mu,k) \neq ([1,2*,3], 4)$ when restricting to regular shears. This suggest several milestones towards proving \cref{conj:squareTiledSurfacesConnectedComponents}, that isolate different points of failure of our proof. We present them ranked by increasing (supposed) difficulty: The first question is whether the quadratic hyperelliptic components are also connected by cylinder shears.

— Reconfiguration of square-tiled surfaces  (2501.15978 - Delecroix et al., 27 Jan 2025) in Section 6, Subsection “Equivalence via cylinder shears”