Reconfiguration of square-tiled surfaces

TL;DR

This paper studies a reconfiguration problem on square-tiled surfaces, showing that certain transformations connect surfaces within hyperelliptic components, and conjectures a broader correspondence with moduli space components.

Contribution

It introduces a new reconfiguration move for square-tiled surfaces and proves its connectivity in hyperelliptic components, linking combinatorial moves to moduli space topology.

Findings

Connected hyperelliptic square-tiled surfaces with O(g) moves

Conjecture relating reconfiguration components to moduli space

Proven connectivity in specific surface classes

Abstract

We consider a combinatorial reconfiguration problem on a subclass of quadrangulations of surfaces called square-tiled surfaces. Our elementary move is a shear in a cylinder that corresponds to a well-chosen sequence of diagonal flips that preserves the square-tiled properties. We conjecture that the connected components of this reconfiguration problem are in bijection with the connected components of the moduli space of quadratic differentials. We prove that the conjecture holds in the so-called hyperelliptic components of Abelian square-tiled surfaces. More precisely, we show that any two such square-tiled surfaces of genus $g$ can be connected by $O(g)$ powers of cylinder shears.