Reconfiguration of square-tiled surfaces

TL;DR

This work studies the reconfiguration problem for square-tiled surfaces under cylinder shears, positing a bijection between cylinder-shear connected components and moduli-space components of quadratic differentials. It proves this conjecture for hyperelliptic components of Abelian square-tiled surfaces by showing any two genus $g$ examples can be connected via $O(g)$ cylinder-shear operations, using a reduction to path-like configurations and a suite of moves (glue/cut, decoration exchange) on decorated plane trees. The approach combines combinatorial encodings (tricoloured involutions, weighted stable graphs) with geometric insight from Abelian/Quadratic differential theory and hyperelliptic quotients, yielding a concrete connectivity bound and a roadmap to extend to other strata. The results illuminate the interplay between discrete reconfiguration dynamics and the geometry of moduli spaces, and raise natural questions about quadratic hyperelliptic components and the full sphere/stratum landscape, as well as potential links to mixing and non-local reconfiguration phenomena such as Kempe changes.

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.

Figures (25)

• Figure 1: Two combinatorial maps that differ by one flip. The flipped edge is dashed. This flip preserves the degree of the faces.
• Figure 2: Diagonal flips do not preserve square-tiled surfaces.
• Figure 3: A horizontal cylinder shear.
• Figure 4: Two square-tiled surfaces in $\operatorname{ST}([4^2])$ (genus $g=2$) that are not connected under cylinder shears.
• Figure 5: Labellings of the two square-tiled surfaces from \ref{['fig:AbQuadSameStratum']}.

Theorems & Definitions (63)

• Theorem 1.1: Disarlo-Parlier DisarloParlier2019
• Theorem 1.2: Budzinski2017CaraceniStauffer2020
• Theorem 1.3: KontsevichZorich2003Lanneau2008
• Conjecture 1.3: Generalizing Conjecture 4 in BuchinEtAl
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2