An arc graph distance formula for the flip graph

TL;DR

This paper establishes a Masur-Minsky style distance formula for the flip graph of triangulations, expressing the distance as a sum of projection distances into arc graphs of subsurfaces, advancing understanding of flip graph geometry.

Contribution

It introduces a novel distance formula for flip graphs based on arc graph projections, extending Masur-Minsky techniques to this setting.

Findings

Proves a Masur-Minsky style distance formula for flip graphs.

Expresses flip graph distance as a sum over subsurface arc graph projections.

Provides a new tool for analyzing triangulation spaces.

Abstract

Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S.