Algebraic intersection for translation sufaces in a family of Teichm\H{u}ller disks

TL;DR

This paper introduces a hyperbolic-geometric method to compute the algebraic intersection ratio KVol for a family of square-tiled translation surfaces within Teichmüller disks, revealing new geometric insights.

Contribution

It provides a novel hyperbolic-geometric construction to calculate KVol for translation surfaces in Teichmüller disks, advancing understanding of their intersection properties.

Findings

Explicit computation of KVol for certain square-tiled surfaces

New geometric interpretation of algebraic intersection ratios

Enhanced understanding of translation surface geometry

Abstract

The setting is a square-tiled surface X. We study the quantity KVol, defined as the supremum over all pairs of closed curves, of their algebraic intersection divided by the product of their length, times the volume of X (so as to make it scaling-invariant). We give a hyperbolic-geometric construction to compute KVol in a family of Teichm\H{u}ller disks of square-tiled surfaces.