Connected components of the SL2(Z) reconfiguration graph

Classify the connected components of the reconfiguration graph on square-tiled surfaces generated by the simultaneous horizontal and vertical cylinder-shear actions corresponding to the generators of $SL_2(\mathbb Z)$.

Background

The paper notes that the natural $SL_2(\mathbb Z)$ action on square-tiled surfaces can be expressed through simultaneous cylinder shears in all horizontal or all vertical cylinders. Because these global moves generate a more restrictive reconfiguration relation than arbitrary cylinder shears, the resulting connected components are finer.

Beyond the flat torus stratum $H(\emptyset)$ and partial results in genus two and Prym loci, the classification is unresolved. The authors state that even a conjectural general classification is unavailable.